Recent zbMATH articles in MSC 46https://zbmath.org/atom/cc/462021-11-25T18:46:10.358925ZUnknown authorWerkzeugBook review of: S. Cobzaş et al., Lipschitz functions.https://zbmath.org/1472.000202021-11-25T18:46:10.358925Z"Kunzinger, M."https://zbmath.org/authors/?q=ai:kunzinger.michaelReview of [Zbl 1431.26002].Computable copies of \(\ell^{p^1}\)https://zbmath.org/1472.030422021-11-25T18:46:10.358925Z"McNicholl, Timothy H."https://zbmath.org/authors/?q=ai:mcnicholl.timothy-hSummary: Suppose \(p\) is a computable real so that \(p\geqslant 1\). It is shown that the halting set can compute a surjective linear isometry between any two computable copies of \(\ell^p\) . It is also shown that this result is optimal in that when \(p\neq 2\) there are two computable copies of \(\ell^p\) with the property that any oracle that computes a linear isometry of one onto the other must also compute the halting set. Thus, \(\ell^p\) is \(\Delta_2^0\)-categorical and is computably categorical if and only if \(p = 2\) . It is also demonstrated that there is a computably categorical Banach space that is not a Hilbert space. These results hold in both the real and complex case.The Ramsey property for operator spaces and noncommutative Choquet simpliceshttps://zbmath.org/1472.051452021-11-25T18:46:10.358925Z"Bartošová, Dana"https://zbmath.org/authors/?q=ai:bartosova.dana"López-Abad, Jordi"https://zbmath.org/authors/?q=ai:lopez-abad.jordi"Lupini, Martino"https://zbmath.org/authors/?q=ai:lupini.martino"Mbombo, Brice"https://zbmath.org/authors/?q=ai:mbombo.brice-rSummary: The noncommutative Gurarij space \(\mathbb{NG} \), initially defined by \textit{T. Oikhberg} [Arch. Math. 86, No. 4, 356--364 (2006; Zbl 1119.46045)], is a canonical object in the theory of operator spaces. As the Fraïssé limit of the class of finite-dimensional nuclear operator spaces, it can be seen as the noncommutative analogue of the classical Gurarij Banach space. In this paper, we prove that the automorphism group of \(\mathbb{NG}\) is extremely amenable, i.e. any of its actions on compact spaces has a fixed point. The proof relies on the Dual Ramsey Theorem, and a version of the Kechris-Pestov-Todorcevic correspondence in the setting of operator spaces.
Recent work of \textit{K. R. Davidson} and \textit{M. Kennedy} [``Noncommutative Choquet theory'', Preprint, \url{arXiv:1905.08436}], building on previous work of Arveson, Effros, Farenick, Webster, and Winkler, among others, shows that nuclear operator systems can be seen as the noncommutative analogue of Choquet simplices. The analogue of the Poulsen simplex in this context is the matrix state space \(\mathbb{NP}\) of the Fraïssé limit \(A(\mathbb{NP})\) of the class of finite-dimensional nuclear operator systems. We show that the canonical action of the automorphism group of \(\mathbb{NP}\) on the compact set \(\mathbb{NP}_1\) of unital linear functionals on \(A( \mathbb{NP})\) is minimal and it factors onto any minimal action, whence providing a description of the universal minimal flow of \(\operatorname{Aut} (\mathbb{NP})\).Sheaf representations and locality of Riesz spaces with order unithttps://zbmath.org/1472.060142021-11-25T18:46:10.358925Z"Di Nola, Antonio"https://zbmath.org/authors/?q=ai:di-nola.antonio"Lenzi, Giacomo"https://zbmath.org/authors/?q=ai:lenzi.giacomo"Spada, Luca"https://zbmath.org/authors/?q=ai:spada.lucaAuthors' abstract: We present an algebraic study of Riesz spaces (= real vector lattices) with a (strong) order unit. We exploit a categorical equivalence between those structures and a variety of algebras called RMV-algebras. We prove two different sheaf representations for Riesz spaces with order unit: the first represents them as sheaves of linearly ordered Riesz spaces over a spectral space, the second represent them as sheaves of ``local'' Riesz spaces over a compact Hausdorff space. Motivated by the latter representation we study the class of local RMV-algebras. We study the algebraic properties of local RMV-algebra and provide a characterisation of them as special retracts of the real interval \([0,1]\). Finally, we prove that the category of local RMV-algebras is equivalent to the category of all Riesz spaces.Local extension property for finite height spaceshttps://zbmath.org/1472.060172021-11-25T18:46:10.358925Z"Correa, Claudia"https://zbmath.org/authors/?q=ai:correa.claudia"Tausk, Daniel V."https://zbmath.org/authors/?q=ai:tausk.daniel-victorSummary: We introduce a new technique for the study of the local extension property (${\mathop \mathrm{LEP}}$) for boolean algebras and we use it to show that the clopen algebra of every compact Hausdorff space $K$ of finite height has $\mathop \mathrm{LEP}$. This implies, under appropriate additional assumptions on $K$ and Martin's Axiom, that every twisted sum of $c_0$ and $C(K)$ is trivial, generalizing a recent result by \textit{W. Marciszewski} and \textit{G. Plebanek} [J. Funct. Anal. 274, No. 5, 1491--1529 (2018; Zbl 1390.46016)].Topologies on abelian lattice ordered groups induced by a positive filter and completenesshttps://zbmath.org/1472.060202021-11-25T18:46:10.358925Z"Jordan, Francis"https://zbmath.org/authors/?q=ai:jordan.francis"Pajoohesh, Homeira"https://zbmath.org/authors/?q=ai:pajoohesh.homeiraSummary: We consider topologies on an abelian lattice ordered group that are determined by the absolute value and a positive filter. We show that the topological completions of these objects are also determined by the absolute value and a positive filter. We investigate the connection between the topological completion of such objects and the Dedekind-MacNeille completion of the underlying lattice ordered group. We consider the preservation of completeness for such topologies with respect to homomorphisms of lattice ordered groups. Finally, we show that topologies defined in terms of absolute value and a positive filter on the space \(C(X)\) of all real-valued continuous functions defined on a completely regular topological space \(X\) are always complete.Spectral spaces of countable abelian lattice-ordered groupshttps://zbmath.org/1472.060242021-11-25T18:46:10.358925Z"Wehrung, Friedrich"https://zbmath.org/authors/?q=ai:wehrung.friedrichSummary: It is well known that the \textit{\( \ell \)-spectrum} of an Abelian \( \ell \)-group, defined as the set of all its prime \( \ell \)-ideals with the hull-kernel topology, is a completely normal generalized spectral space. We establish the following converse of this result.
Theorem. Every second countable, completely normal generalized spectral space is homeomorphic to the \( \ell \)-spectrum of some Abelian \( \ell \)-group.
We obtain this result by proving that a countable distributive lattice \( D\) with zero is isomorphic to the Stone dual of some \( \ell \)-spectrum (we say that \( D\) is \textit{\( \ell \)-representable}) iff for all \( a,b\in D\) there are \( x,y\in D\) such that \( a\vee b=a\vee y=b\vee x\) and \( x\wedge y=0\). On the other hand, we construct a non-\( \ell \)-representable bounded distributive lattice, of cardinality \( \aleph _1\), with an \( \ell \)-representable countable \( {\mathscr {L}}_{\infty ,\omega }\)-elementary sublattice. In
particular, there is no characterization, of the class of all \( \ell \)-representable distributive lattices, by any class of \( {\mathscr {L}}_{\infty ,\omega }\) sentences.On a relation between density measures and a certain flowhttps://zbmath.org/1472.110512021-11-25T18:46:10.358925Z"Kunisada, Ryoichi"https://zbmath.org/authors/?q=ai:kunisada.ryoichiSummary: We study extensions of the asymptotic density to a finitely additive measure defined on all subsets of natural numbers. Such measures are called density measures. We consider a class of density measures constructed from free ultrafilters on natural numbers and investigate absolute continuity and singularity for those density measures. In particular, for any pair of such density measures we prove necessary and sufficient conditions that one is absolutely continuous with respect to the other and that they are singular. Also, we prove similar results for weak absolute continuity and strong singularity. These results are formulated in terms of topological dynamics.A positivity conjecture related to the Riemann zeta functionhttps://zbmath.org/1472.112272021-11-25T18:46:10.358925Z"Bellemare, Hugues"https://zbmath.org/authors/?q=ai:bellemare.hugues"Langlois, Yves"https://zbmath.org/authors/?q=ai:langlois.yves"Ransford, Thomas"https://zbmath.org/authors/?q=ai:ransford.thomas-jSummary: According to two remarkable theorems of \textit{B. Nyman} [On the one-dimensional translation group and semi-group in certain function spaces. Uppsala: Appelbergs Boktryckeri AB 55 S (1950; Zbl 0037.35401)] and \textit{L. Báez-Duarte} [Atti Accad. Naz. Lincei, Cl. Sci. Fis. Mat. Nat., IX. Ser., Rend. Lincei, Mat. Appl. 14, No. 1, 5--11 (2003; Zbl 1097.11041)], the Riemann hypothesis is equivalent to a simply stated criterion concerning least-squares approximation. In carrying out computations related to this criterion, we have observed a curious phenomenon: for no apparent reason, at least the first billion entries of a certain infinite triangular matrix associated with the Riemann zeta function are all positive. In this article, we describe the background leading to this observation, and make a conjecture.The set of separable states has no finite semidefinite representation except in dimension \(3\times 2\)https://zbmath.org/1472.140662021-11-25T18:46:10.358925Z"Fawzi, Hamza"https://zbmath.org/authors/?q=ai:fawzi.hamzaGiven integers \(n \ge m,\) let \(\mathrm{Sep}(n, m)\) be the set of {\em separable states} on the Hilbert space \(\mathbb{C}^n \otimes \mathbb{C}^m,\) i.e., \[\mathrm{Sep}(n, m) := \mathbf{conv}\{x x^\dagger \otimes y y^\dagger \ : \ x \in \mathbb{C}^n, |x| = 1, y \in \mathbb{C}^m, |y| = 1\}.\] Here \(x^\dagger\) indicates conjugate transpose, \(|x|^2 := x^\dagger x\) and \(\mathbf{conv}\) denotes the convex hull.
We say that a convex set \(C \subset \mathbb{R}^d\) has a {\em semidefinite representation} (of size \(r\)) if it can be expressed as \(C = \pi(S),\) where \(\pi \colon \mathbb{R}^D \to \mathbb{R}^d\) is a linear map and \(S \subset \mathbb{R}^D\) is a convex set defined using a linear matrix inequality \[S = \{w \in \mathbb{R}^D \ : \ M_0 + w_1M_1 + \cdots + w_DM_D \succeq 0\}\] where \(M_0, \ldots, M_D\) are Hermitian matrices of size \(r \times r.\)
It is known, from the earlier work of [\textit{S. L. Woronowicz}, Rep. Math. Phys. 10, 165--183 (1976; Zbl 0347.46063)] that for \(n + m \le 5,\) the set \(\mathrm{Sep}(n, m)\) is just the set of states which have a positive partial transpose, and hence it has a semidefinite representation.
In the paper under review, the author shows that for \(n + m > 5,\) the set \(\mathrm{Sep}(n, m)\) has no semidefinite representation, and so this provides a new counterexample to the Helton-Nie conjecture [\textit{J. W. Helton} and \textit{J. Nie}, SIAM J. Optim. 20, No. 2, 759--791 (2009; Zbl 1190.14058)], which was recently disproved by \textit{C. Scheiderer} [SIAM J. Appl. Algebra Geom. 2, No. 1, 26--44 (2018; Zbl 1391.90462)].
The paper is very clear, well written and quite interesting.On the rank and the approximation of symmetric tensorshttps://zbmath.org/1472.150362021-11-25T18:46:10.358925Z"Rodríguez, Jorge Tomás"https://zbmath.org/authors/?q=ai:rodriguez.jorge-tomasSummary: In this work we study different notions of ranks and approximation of tensors. We consider the tensor rank, the nuclear rank and we introduce the notion of symmetric decomposable rank, a notion of rank defined only on symmetric tensors. We show that when approximating symmetric tensors, using the symmetric decomposable rank has some significant advantages over the tensor rank and the nuclear rank.On the stability of left \(\delta \)-centralizers on Banach Lie triple systemshttps://zbmath.org/1472.170152021-11-25T18:46:10.358925Z"Ghobadipour, Norouz"https://zbmath.org/authors/?q=ai:ghobadipour.norouz"Sepasian, Ali Reza"https://zbmath.org/authors/?q=ai:sepasian.ali-rezaSummary: In this paper under a condition, we prove that every Jordan left \(\delta \)-centralizer on a Lie triple system is a left \(\delta \)-centralizer. Moreover, we use a fixed point method to prove the generalized Hyers-Ulam-Rassias stability associated with the Pexiderized Cauchy-Jensen type functional equation \[rf\left(\frac{x+y}{r}\right)+sg\left(\frac{x-y}{s}\right)=2h(x),\] for \(r,s \in \mathbb R \setminus \{0\}\) in Banach Lie triple systems.Description of 2-local and local derivations on some Lie rings of skew-adjoint matriceshttps://zbmath.org/1472.170622021-11-25T18:46:10.358925Z"Ayupov, Sh. A."https://zbmath.org/authors/?q=ai:ayupov.sh-a"Arzikulov, F. N."https://zbmath.org/authors/?q=ai:arzikulov.farhodjon-nematjonovichSummary: In the present paper, we prove that every 2-local inner derivation on the Lie ring of skew-symmetric matrices over a commutative ring is an inner derivation. We also apply our technique to various Lie algebras of infinite-dimensional skew-adjoint matrix-valued maps on a set and prove that every 2-local spatial derivation on such algebras is a spatial derivation. A similar technique is applied to the same Lie algebras and proved that every local spatial derivation on such algebras is a spatial derivation.A note on a certain Baum-Connes map for inverse semigroupshttps://zbmath.org/1472.190032021-11-25T18:46:10.358925Z"Burgstaller, Bernhard"https://zbmath.org/authors/?q=ai:burgstaller.bernhardIn operator \(K\)-theory, the Baum-Connes conjecture [\textit{A. Valette}, Lectures in mathematics. Basel: Birkhäuser (2002; Zbl 1136.58013)] suggests a link between the \(K\)-theory of the reduced \(C^*\)-algebra of a locally compact group and the \(K\)-homology of the classifying space of proper actions of that group. The conjecture sets up a correspondence between different areas of mathematics, in particular topology and geometry, and implies famous conjectures as consequences (e.g., the the Novikov conjecture).
The aim of this paper is formulating an analogous conjecture not for groups but for (discrete) inverse semigroups. A semigroup is a set equipped with an associative binary operation, the adjective ``inverse'' refers to the existence of a \textit{unique inverse} in the following sense: for each \(x\) in the semigroup \(S\), there exists \(y\in S\) such that \(xyx=x\) and \(yxy=y\). In such a semigroup \(S\), the subset of idempotents \(E\subseteq S\) form a \textit{semilattice} (i.e., idempotents commute).
The spectrum of the abelian \(C^*\)-algebra \(C^*(E)\) is the space of \textit{characters} \(X\). Any \(e\in E\) defines a character \(\epsilon_e\in X\) by setting \(\epsilon_e(f)=1_{\{f\geq e\}}\). An \(S\)-\(C^*\)-algebra \(A\) is by definition equipped with a map \(\alpha\colon S\to \mathrm{End}(A)\) such that \(\alpha_e(a)b=a\alpha_e(b)\) for all \(a,b\in A\) and all idempotents \(e\in S\). It is in particular a \(C_0(X)\)-algebra, namely it gives rise to an upper-semicontinuous \(C^*\)-bundle over \(X\).
The strategy is based on a formulation (obtained in [\textit{R. Meyer} and \textit{R. Nest}, Topology 45, No. 2, 209--259 (2006; Zbl 1092.19004)]) of the Baum-Connes conjecture based on Verdier localization of the triangulated equivariant bivariant Kasparov category \(K\!K^G\) (where \(G\) is a group).
Roughly speaking, Meyer and Nest [loc. cit.] show that the conjecture (for a given coefficent \(G\)-\(C^*\)-algebra \(A\)) asks that a certain map \(D_A\colon P(A)\to A\) is an isomorphism, where \((P(A),D_A)\) is obtained via the previously mentioned localization at the subcategory of \textit{weakly contractible} objects, i.e., objects \(B\in K\!K^G\) such that \(\mathrm{Res}^H_G(B)\cong 0\) for any compact subgroup \(H\subseteq G\).
\textit{R. Meyer} [Tbil. Math. J. 1, 165--210 (2008; Zbl 1161.18301)] shows that \((P(A),D_A)\) can be constructed if one can prove an adjoint situation
\[
K\!K^G(\mathrm{Ind}_H^G(A),B)\cong K\!K^H(A,\mathrm{Res}^H_G(B)).
\]
In view of this, for a given finite inverse subsemigroup \(H\subseteq S\), the author defines \textit{induction} and \textit{restriction} functors for \(C^*\)-algebras equipped with semigroup actions.
The induction functor has been previously defined by the same author in [Semigroup Forum 100, No. 1, 141--152 (2020; Zbl 07169448)], and satisfies an analogue of Green's imprimitivity theorem, as is to be expected. The author also previously attempted to define a restriction functor in [\textit{B. Burgstaller}, Aust. J. Math. Anal. Appl. 17, No. 2, Article No. 1, 22 p. (2020; Zbl 1463.19002)], however the functor defined in the present paper is different and more well-behaved. The restriction functor in this paper is defined as \(\mathrm{Res}^H_S(B)=\oplus_{e\in E_H} A_{\epsilon_e}\), where \(E_H\) is the idempotent set of \(H\).
In particular, the author proves the adjoint situation above for \(H\subseteq S\), under the restriction that \(B\) is a so-called \textit{fibered} algebra, i.e., a \(G\)-algebra of the form \(\oplus_{e\in E} A_{\epsilon_e}\). Hence we get a Baum-Connes type conjecture for fibered coefficient algebras.
This is a severe limitation, in particular it prevents from formulating a meaningful conjecture for the basic crossed product \(\mathbb{C}\rtimes S\). We should also point out that the result rests upon the triangulated structure of the (full) subcategory generated by objects isomorphic to fibered algebras, something for which the author does not give sufficient details (more precisely, the author makes use of an adaptation of the main theorem of [\textit{R. Meyer}, \(K\)-Theory 21, No. 3, 201--228 (2000; Zbl 0982.19004)] to the case of semigroups, which is left unproved).On the classification of group actions on \(C^*\)-algebras up to equivariant \(KK\)-equivalencehttps://zbmath.org/1472.190042021-11-25T18:46:10.358925Z"Meyer, Ralf"https://zbmath.org/authors/?q=ai:meyer.ralf.1The paper deals with the classification of group actions on \(C^*\)-algebras up to equivariant \(KK\)-equivalence. Let \(G\) be a second countable, locally compact group. It is shown that any group action is equivariantly \(KK\)-equivalent to an action on a simple, purely infinite \(C^*\)-algebra. If \(A\) is a Kirchberg algebra, it is shown that the \(KK^G\)-equivalence classes of actions of a torsion-free amenable group \(G\) on \(A\) are in one-to-one correspondence with the isomorphism classes of principal \(\operatorname{Aut}(A)\)-bundles over the classifying space \(BG\). For a cyclic group of prime order, its actions (not necessarily satisfying the Rokhlin property) are classified up to equivariant \(KK\)-equivalence using the Köhler invariant, in particular, the classification of \(\mathbb Z/p\) actions on stabilised Cuntz algebras in the equivariant bootstrap class is described.Strong Novikov conjecture for low degree cohomology and exotic group \(\mathrm{C}^\ast\)-algebrashttps://zbmath.org/1472.190052021-11-25T18:46:10.358925Z"Antonini, Paolo"https://zbmath.org/authors/?q=ai:antonini.paolo"Buss, Alcides"https://zbmath.org/authors/?q=ai:buss.alcides"Engel, Alexander"https://zbmath.org/authors/?q=ai:engel.alexander"Siebenand, Timo"https://zbmath.org/authors/?q=ai:siebenand.timoLet \(G\) be a discrete group, \(\Lambda^*(G) \subset H^*(BG;\mathbb Q)\) the subring generated by the rational cohomology classes of degree at most two, and let \(\mathrm{ch}: K_*(BG) \to H_*(BG;\mathbb Q)\) be the homological Chern character from the \(K\)-homology to the homology of the classifying space \(BG\) of \(G\).
It was shown in [\textit{B. Hanke} and \textit{T. Schick}, Geom. Dedicata 135, 119--127 (2008; Zbl 1149.19006)] that if for \(h\in K_*(BG)\) there exists \(c \in\Lambda^*(G)\) with \(\langle c, \mathrm{ch}(h)\rangle\neq 0\) then \(h\) is not mapped to zero under the assembly map \(K_*(BG) \to K_*(C^*_{\max}(G)) \otimes\mathbb R\).
The paper under review shows that the maximal group \(C^*\)-algebra here can be replaced by a smaller one, namely, by the exotic group \(C^*\)-algebra \(C^*_\epsilon (G)\), which is obtained from the so-called minimal exact and strongly Morita compatible crossed product functor introduced in [\textit{P. Baum} et al., Ann. \(K\)-Theory 1, No. 2, 155--208 (2016; Zbl 1331.46064)].Symmetry of eigenvalues of operators associated with representations of compact quantum groupshttps://zbmath.org/1472.201162021-11-25T18:46:10.358925Z"Krajczok, Jacek"https://zbmath.org/authors/?q=ai:krajczok.jacekSummary: We ask whether for a given unitary representation $U$ of a compact quantum group $\mathbb G$ the associated operator $\rho_{U}\in\operatorname{Mor}(U,U^{\text{cc}})$ has spectrum invariant under inversion; we then say that $\rho_{U}$ has symmetric eigenvalues. This is not always the case. However, we give an affirmative answer whenever a certain condition on the growth of the dimensions of irreducible subrepresentations of tensor powers of $U$ is imposed. This condition is satisfied whenever $\widehat{\mathbb G}$ is of subexponential growth.An algebraic approach to the Weyl groupoidhttps://zbmath.org/1472.220012021-11-25T18:46:10.358925Z"Bice, Tristan"https://zbmath.org/authors/?q=ai:bice.tristan-matthewThe Kumjian-Renault Weyl groupoid construction and the Lawson-Lenz version of Exel's tight groupoid construction are unified by utilising only a weak algebraic fragment of the \(C^*\)-algebra structure, that is, its *-semigroup reduct. The author also prove that local compactness is still valid in general classes of *-rings.Spectrality of generalized Sierpinski-type self-affine measureshttps://zbmath.org/1472.280072021-11-25T18:46:10.358925Z"Liu, Jing-Cheng"https://zbmath.org/authors/?q=ai:liu.jingcheng"Zhang, Ying"https://zbmath.org/authors/?q=ai:zhang.ying.3|zhang.ying.4|zhang.ying|zhang.ying.1|zhang.ying.5|zhang.ying.2"Wang, Zhi-Yong"https://zbmath.org/authors/?q=ai:wang.zhiyong.2|wang.zhiyong.1"Chen, Ming-Liang"https://zbmath.org/authors/?q=ai:chen.ming-liangSummary: In this work, we study the spectral property of generalized Sierpinski-type self-affine measures \(\mu_{M,D}\) on \(\mathbb{R}^2\) generated by an expanding integer matrix \(M\in M_2(\mathbb{Z})\) with \(\det(M)\in 3\mathbb{Z}\) and a non-collinear integer digit set \(D=\{(0,0)^t,(\alpha_1,\alpha_2)^t,(\beta_1,\beta_2)^t\}\) with \(\alpha_1\beta_2-\alpha_2\beta_1\in 3\mathbb{Z}\). We give the sufficient and necessary conditions for \(\mu_{M,D}\) to be a spectral measure, i.e., there exists a countable subset \(\Lambda\subset \mathbb{R}^2\) such that \(E(\Lambda)=\{e^{2\pi i\langle\lambda,x \rangle}:\lambda\in\Lambda\}\) forms an orthonormal basis for \(L^2(\mu_{M,D})\). This completely settles the spectrality of the self-affine measure \(\mu_{M,D}\).Spectrality of a class of self-affine measures and related digit setshttps://zbmath.org/1472.280132021-11-25T18:46:10.358925Z"Yang, Ming-Shu"https://zbmath.org/authors/?q=ai:yang.ming-shuSummary: This work investigates the spectrality of a self-affine measure \(\mu_{M,D}\) and the related digit set \(D\) in the case when \(|\mathrm{det}(M)|=p^{\alpha}\) is a prime power and \(|D|=p\) is a prime, where \(\alpha\in{\mathbb{N}}\), and \(\mu_{M,D}\) is generated by an expanding matrix \(M\in M_n({\mathbb{Z}})\) and a digit set \(D\subset\mathbb{Z}^n\) of cardinality |\(D\)|. We obtain that \(\mu_{M,D}\) is a spectral measure and \(D\) is a spectral set if one nonzero element in \(D\) satisfies certain mild conditions. This is based on the property of vanishing sums of roots of unity and a residue system in number theory. The result here extends the corresponding known results and provides some supportive evidence for a conjecture of Dutkay, Han, and Jorgensen.Weak estimates for the maximal and Riesz potential operators in central Herz-Morrey spaces on the unit ballhttps://zbmath.org/1472.310082021-11-25T18:46:10.358925Z"Mizuta, Yoshihiro"https://zbmath.org/authors/?q=ai:mizuta.yoshihiro"Ohno, Takao"https://zbmath.org/authors/?q=ai:ohno.takao"Shimomura, Tetsu"https://zbmath.org/authors/?q=ai:shimomura.tetsuThis nice paper introduces the weak central Herz-Morrey spaces \(WH^{p(\cdot),q,\omega}(\mathbf B)\) and \(WH^{p^\ast(\cdot),q,\omega}(\mathbf B)\) with \( p^\ast(x) = p(x)N/(N-\alpha p(x))\) on the Euclidean unit ball \(\mathbf B\) and shows the boundedness of the generalized maximal operator \(M_\beta\) and the Riesz potential operator \(I_\alpha\) from the non-homogeneous central Herz-Morrey space \(H^{p(\cdot),q,\omega}(\mathbf B)\) to \(WH^{p(\cdot),q,\omega}(\mathbf B)\) (Theorem 3.10) and \(WH^{p^\ast(\cdot),q,\omega}(\mathbf B)\) (Theorem 4.1), respectively.A uniqueness result for functions with zero fine gradient on quasiconnected and finely connected setshttps://zbmath.org/1472.310132021-11-25T18:46:10.358925Z"Björn, Anders"https://zbmath.org/authors/?q=ai:bjorn.anders"Björn, Jana"https://zbmath.org/authors/?q=ai:bjorn.janaThis article establishes that every Sobolev function in \(W^{1,p}_{\mathrm{loc}}(U)\) on a \(p\)-quasiopen set \(U\subset \mathbb{R}^n\) with almost everywhere vanishing \(p\)-fine gradient is almost everywhere constant if and only if \(U\) is \(p\)-quasiconnected. The approach relies on the theory of Newtonian Sobolev spaces on metric measure spaces.Twin semigroups and delay equationshttps://zbmath.org/1472.341202021-11-25T18:46:10.358925Z"Diekmann, O."https://zbmath.org/authors/?q=ai:diekmann.odo"Verduyn Lunel, S. M."https://zbmath.org/authors/?q=ai:verduyn-lunel.sjoerd-mAuthors' abstract: In the standard theory of delay equations, the fundamental solution does not `live' in the state space. To eliminate this age-old anomaly, we enlarge the state space. As a consequence, we lose the strong continuity of the solution operators and this, in turn, has as a consequence that the Riemann integral no longer suffices for giving meaning to the variation-of-constants formula. To compensate, we develop the Stieltjes-Pettis integral in the setting of a norming dual pair of spaces. Part I provides general theory, Part II deals with ``retarded'' equations, and in Part III we show how the Stieltjes integral enables incorporation of unbounded perturbations corresponding to neutral delay equations.\(L^p\)-versions of generalized Korn inequalities for incompatible tensor fields in arbitrary dimensions with \(p\)-integrable exterior derivativehttps://zbmath.org/1472.350152021-11-25T18:46:10.358925Z"Lewintan, Peter"https://zbmath.org/authors/?q=ai:lewintan.peter"Neff, Patrizio"https://zbmath.org/authors/?q=ai:neff.patrizioSummary: For \(n\geq 2\) and \(1<p<\infty\) we prove an \(L^p\)-version of the generalized Korn-type inequality for incompatible, \(p\)-integrable tensor fields \(P:\Omega\rightarrow\mathbb{R}^{n\times n}\) having \(p\)-integrable generalized \(\underline{\mathrm{Curl}}\) and generalized vanishing tangential trace \(P\tau_l=0\) on \(\partial\Omega\), denoting by \(\{\tau_l\}_{l=1,\dots,n-1}\) a moving tangent frame on \(\partial\Omega\), more precisely we have:
\[
\Vert P\Vert_{L^p(\Omega,\mathbb{R}^{n\times n})}\leq c\left(\left\Vert \operatorname{sym}P\right\Vert_{L^p(\Omega,\mathbb{R}^{n\times n})}+\left\Vert\underline{\mathrm{Curl}}\,P \right\Vert_{L^p(\Omega,(\mathfrak{so}(n))^n)}\right),
\]
where the generalized \(\underline{\mathrm{Curl}}\) is given by \((\underline{\mathrm{Curl}}\,P)_{ijk}:=\partial_iP_{kj}-\partial_jP_{ki}\) and \(c=c(n,p,\Omega )>0\).Modular maximal estimates of Schrödinger equationshttps://zbmath.org/1472.350702021-11-25T18:46:10.358925Z"Ho, Kwok-Pun"https://zbmath.org/authors/?q=ai:ho.kwok-punSummary: This paper offers the maximal estimates of the solutions of some initial value problems on modular spaces. Our results include the estimates for the solutions of Schrödinger equation.On the strong maximum principle for a fractional Laplacianhttps://zbmath.org/1472.350732021-11-25T18:46:10.358925Z"Trong, Nguyen Ngoc"https://zbmath.org/authors/?q=ai:trong.nguyen-ngoc"Tan, Do Duc"https://zbmath.org/authors/?q=ai:tan.do-duc"Thanh, Bui Le Trong"https://zbmath.org/authors/?q=ai:thanh.bui-le-trongLet \(\Omega\) be a bounded open connected set in \(\mathbb{R}^n\) with Lipschitz boundary, \(s\in (\frac{1}{2},1)\), \(\{e_k\}\) be the sequence of eigenfunctions of the Laplace operator \(-\Delta: H_0^2(\Omega)\rightarrow L^2(\Omega)\), and \(\{\lambda_k\}\) be the sequence of corresponding eigenvalues.
The authors establish a strong maximum principle for the spectral Dirichlet Laplacian \((-\Delta)^s\), defined by \[(-\Delta)^su=\sum_{k\in \mathbb{N}}\lambda_{k}^s\langle e_k,u\rangle_{L^2(\Omega)}e_k\] for each \(u=L^2(\Omega)\) such that \(\sum_{k\in \mathbb{N}}u_k^2\lambda_k^2<\infty\), where \(u_k\) is the component of \(u\) along \(e_k\).
In particular, the authors prove that every nonnegative function \(u\in L^1(\Omega)\) such that \((-\Delta)^su\) is a Radon measure on \(\Omega\) is almost everywhere equal to a quasi continuous function \(\tilde{u}\), and if \(\tilde{u}=0\) on a subset of \(\Omega\) with positive \(H^s\)-capacity and \(u\) satisfies \[(-\Delta)^su+au\geq 0 \ \ \ \text{a.e. in} \ \ \Omega,\] for some nonnegative function \(a\in L^1(\Omega)\), then \(u=0\) in \(\Omega\).
Ingredients of the proof are a fractional version of the Poincaré's inequality and truncated functions.Best regularity for a Schrödinger type equation with non smooth data and interpolation spaceshttps://zbmath.org/1472.351132021-11-25T18:46:10.358925Z"Rakotoson, Jean Michel"https://zbmath.org/authors/?q=ai:rakotoson.jean-michelSummary: Given a vector field \(U(x)\) and a nonnegative potential \(V(x)\) on a smooth open bounded set \(\Omega\) of \(\mathbb{R}^n\), we shall discuss some regularity results for the following equation \[-\Delta\omega +U\cdot\nabla\omega+V\omega=f\quad \text{ in }\Omega\tag{0.1}\] whenever \(\delta f\) is a bounded Radon measure with \(\delta(x)\) is the distance between \(x\) and the boundary \(\delta\Omega\).
For the entire collection see [Zbl 1448.65007].Global regularity estimates for non-divergence elliptic equations on weighted variable Lebesgue spaceshttps://zbmath.org/1472.351252021-11-25T18:46:10.358925Z"Bui, The Quan"https://zbmath.org/authors/?q=ai:bui.the-quan"Bui, The Anh"https://zbmath.org/authors/?q=ai:the-anh-bui."Duong, Xuan Thinh"https://zbmath.org/authors/?q=ai:duong.xuan-thinhCorrigendum to: ``Elliptic problems with growth in nonreflexive Orlicz spaces and with measure or \(L^1\) data''https://zbmath.org/1472.351622021-11-25T18:46:10.358925Z"Chlebicka, Iwona"https://zbmath.org/authors/?q=ai:chlebicka.iwona"Giannetti, Flavia"https://zbmath.org/authors/?q=ai:giannetti.flavia"Zatorska-Goldstein, Anna"https://zbmath.org/authors/?q=ai:zatorska-goldstein.annaSummary: The authors would like to correct an error in the proof of uniqueness in their paper [ibid. 479, No. 1, 185--213 (2019; Zbl 1433.35086)].Existence of solution for a class of heat equation with double criticalityhttps://zbmath.org/1472.352272021-11-25T18:46:10.358925Z"Alves, Claudianor O."https://zbmath.org/authors/?q=ai:alves.claudianor-oliveira"Boudjeriou, Tahir"https://zbmath.org/authors/?q=ai:boudjeriou.tahirSummary: In this paper, we study the following class of quasilinear heat equations
\[
\begin{cases}
u_t-\Delta_\Phi u=f(x,u) & \text{ in }\Omega,\quad t>0, \\
u=0 & \text{ on }\partial\Omega,\quad t>0, \\
u(x,0)=u_0(x) & \text{ in }\Omega,
\end{cases}
\]
where \(\Delta_\Phi u=\operatorname{div}(\varphi (x,|\nabla\varphi|)\nabla\varphi)\) and \(\Phi(x,s)=\int_0^{|s|}\varphi(x,\sigma)\sigma d\sigma\) is a generalized N-function. We suppose that \(\Omega\subset\mathbb{R}^N\) \((N\geq 2)\) is a smooth bounded domain that contains two open regions \(\Omega_N\) and \(\Omega_p\) with \(\overline{\Omega}_N\cap\overline{\Omega}_p=\emptyset\). Under some appropriate conditions, the global existence will be done by combining the Galerkin approximations with the potential well theory. Moreover, the large-time behavior of the global weak solution is analyzed. The main feature of this paper consists that \(-\Delta_\Phi u\) behaves like \(-\Delta_Nu\) on \(\Omega_N\) and \(-\Delta_pu\) on \(\Omega_p\), while the continuous function \(f:\Omega\times\mathbb{R}\to\mathbb{R}\) behaves like \(e^{\alpha |s|^{\frac{N}{N-1}}}\) on \(\Omega_N\) and \(|s|^{p_\ast-2}s\) on \(\Omega_p\) as \(|s|\to\infty\).Spectral stability estimates of Dirichlet divergence form elliptic operatorshttps://zbmath.org/1472.352542021-11-25T18:46:10.358925Z"Gol'dshtein, Vladimir"https://zbmath.org/authors/?q=ai:goldshtein.vladimir"Pchelintsev, Valerii"https://zbmath.org/authors/?q=ai:pchelintsev.valerii"Ukhlov, Alexander"https://zbmath.org/authors/?q=ai:ukhlov.alexanderThe paper is aimed on applying quasiconformal mappings to spectral stability estimates of the Dirichlet eigenvalues of \(A\)-divergent form elliptic operators
\[
L_{A}=-\text{div} [A(w) \nabla g(w)]\in \widetilde{\Omega}, \quad w|_{\partial \widetilde{\Omega}}=0,
\]
in non-Lipschitz domains \(\widetilde{\Omega} \subset \mathbb{C}\) with \(2 \times 2\) symmetric matrix functions \(A(w)=\left\{a_{kl}(w)\right\}\), \(\textrm{det} A=1\), with measurable entries satisfying the uniform ellipticity condition.
The main results of the article concern to spectral stability estimates in domains that the authors call as \(A\)-quasiconformal \(\beta\)-regularity domains. Namely, a simply connected domain \(\widetilde{\Omega} \subset \mathbb{C}\) is called an \(A\)-quasiconformal \(\beta\)-regular domain about a simply connected domain \({\Omega} \subset \mathbb{C}\) if
\[
\iint\limits_{\widetilde{\Omega}} |J(w, \varphi)|^{1-\beta}~dudv < \infty, \,\,\,\beta>1,
\]
where \(J(w, \varphi)\) is a Jacobian of an \(A\)-quasiconformal mapping \(\varphi: \widetilde{\Omega}\to\Omega\).
The main result of the article states that, if a domain \(\widetilde{\Omega}\) is \(A\)-quasiconformal \(\beta\)-regular about \(\Omega\), then for any \(n\in \mathbb{N}\) the following spectral stability estimates hold:
\[
|\lambda_n[I, \Omega]-\lambda_n[A, \widetilde{\Omega}]| \leq c_n A^2_{\frac{4\beta}{\beta -1},2}(\Omega) \left(|\Omega|^{\frac{1}{2\beta}} + \|J_{\varphi^{-1}}\,|\,L^{\beta}(\Omega)\|^{\frac{1}{2}} \right) \cdot \|1-J_{\varphi^{-1}}^{\frac{1}{2}}\,|\,L^{2}(\Omega)\|,
\]
where \(c_n=\max\left\{\lambda_n^2[A, \Omega], \lambda_n^2[A, \widetilde{\Omega}]\right\}\), \(J_{\varphi^{-1}}\) is a Jacobian of an \(A^{-1}\)-quasiconformal mapping \(\varphi^{-1}:\Omega\to\widetilde{\Omega}\), and
\[
A_{\frac{4\beta}{\beta -1},2}(\Omega) \leq \inf\limits_{p\in \left(\frac{4\beta}{3\beta -1},2\right)} \left(\frac{p-1}{2-p}\right)^{\frac{p-1}{p}} \frac{\left(\sqrt{\pi}\cdot\sqrt[p]{2}\right)^{-1}|\Omega|^{\frac{\beta-1}{4\beta}}}{\sqrt{\Gamma(2/p) \Gamma(3-2/p)}}~~.
\]Fine metrizable convex relaxations of parabolic optimal control problemshttps://zbmath.org/1472.354152021-11-25T18:46:10.358925Z"Roubíček, Tomáš"https://zbmath.org/authors/?q=ai:roubicek.tomasThe paper deals with fine metrizable convex relaxations of parabolic optimal control problems. In particular, a compromising convex compactification is devised. The basic idea consists in combining classical techniques for Young measures with Choquet theory. Therefore, the proposed approach works under classical \(\sigma\)-additive measures and standard sequences. At the same time, it allows for dealing with a wider class of nonlinearities than only affine. The controls \(u\) are valued in the set \(S_p\) of the form \[ S_p = \{ u \in L^p(\Omega;\mathbb{R}^m): u(x) \in B \; \text{for a.a.} \; x \in \Omega\}, \] with \(\Omega \subset \mathbb{R}^d\), \(d \in \mathbb{N}\), \(1 \leq p < +\infty\) and \(B \subset \mathbb{R}^m\) bounded and closed. In addition, some generalization to unbounded domain \(B\) by considering a general \(S_p\) bounded in \(L^p(\Omega;\mathbb{R}^m)\), with \(1 \leq p < \infty\) fixed but not necessarily bounded in \(L^\infty(\Omega;\mathbb{R}^m)\), is also discussed. Finally, an application to optimal control of a system of semilinear parabolic differential equations is presented for the reader convenience, together with other relaxation strategies as well as more general nonlinearities, showing that the finds reported in the paper are useful for practical applications.Compact embedding theorems and a Lions' type lemma for fractional Orlicz-Sobolev spaceshttps://zbmath.org/1472.354422021-11-25T18:46:10.358925Z"Silva, Edcarlos D."https://zbmath.org/authors/?q=ai:da-silva.edcarlos-domingos"Carvalho, M. L."https://zbmath.org/authors/?q=ai:carvalho.marcos-l-m|carvalho.marcos-leandro"de Albuquerque, J. C."https://zbmath.org/authors/?q=ai:de-albuquerque.jose-carlos"Bahrouni, Sabri"https://zbmath.org/authors/?q=ai:bahrouni.sabriSummary: In this paper we are concerned with some abstract results regarding to fractional Orlicz-Sobolev spaces. Precisely, we ensure the compactness embedding for the weighted fractional Orlicz-Sobolev space into the Orlicz spaces, provided the weight is unbounded. We also obtain a version of Lions' ``vanishing'' Lemma for fractional Orlicz-Sobolev spaces, by introducing new techniques to overcome the lack of a suitable interpolation law. Finally, as a product of the abstract results, we use a minimization method over the Nehari manifold to prove the existence of ground state solutions for a class of nonlinear Schrödinger equations, taking into account unbounded or bounded potentials.Infinite order \(\Psi\mathrm{DOs}\): composition with entire functions, new Shubin-Sobolev spaces, and index theoremhttps://zbmath.org/1472.354692021-11-25T18:46:10.358925Z"Pilipović, Stevan"https://zbmath.org/authors/?q=ai:pilipovic.stevan-r"Prangoski, Bojan"https://zbmath.org/authors/?q=ai:prangoski.bojan"Vindas, Jasson"https://zbmath.org/authors/?q=ai:vindas.jassonSummary: We study global regularity and spectral properties of power series of the Weyl quantisation \(a^w\), where \(a(x,\xi)\) is a classical elliptic Shubin polynomial. For a suitable entire function \(P\), we associate two natural infinite order operators to \(a^w\), \(P(a^w)\) and \((P\circ a)^w\), and prove that these operators and their lower order perturbations are globally Gelfand-Shilov regular. They have spectra consisting of real isolated eigenvalues diverging to \(\infty\) for which we find the asymptotic behaviour of their eigenvalue counting function. In the second part of the article, we introduce Shubin-Sobolev type spaces by means of \(f\)-\(\Gamma^{*,\infty}_{A_p,\rho}\)-elliptic symbols, where \(f\) is a function of ultrapolynomial growth and \(\Gamma^{*,\infty}_{A_p,\rho}\) is a class of symbols of infinite order studied in this and our previous papers. We study the regularity properties of these spaces, and show that the pseudo-differential operators under consideration are Fredholm operators on them. Their indices are independent on the order of the Shubin-Sobolev spaces; finally, we show that the index can be expressed via a Fedosov-Hörmander integral formula.Smale spaces from self-similar graph actionshttps://zbmath.org/1472.370112021-11-25T18:46:10.358925Z"Yi, Inhyeop"https://zbmath.org/authors/?q=ai:yi.inhyeopSummary: We show that, for contracting and regular self-similar graph actions, the shift maps on limit spaces are positively expansive local homeomorphisms. From this, we find that limit solenoids of contracting and regular self-similar graph actions are Smale spaces and that the unstable Ruelle algebras of the limit solenoids are strongly Morita equivalent to the Cuntz-Pimsner algebras by \textit{R. Exel} and \textit{E. Pardo} [Semigroup Forum 92, No. 1, 274--303 (2016; Zbl 1353.20040); Adv. Math. 306, 1046--1129 (2017; Zbl 1390.46050)] if self-similar graph actions satisfy the contracting, regular, pseudo free and \(G\)-transitive conditions.On superstability of the Wigner equationhttps://zbmath.org/1472.390492021-11-25T18:46:10.358925Z"Ilišević, Dijana"https://zbmath.org/authors/?q=ai:ilisevic.dijana"Turnšek, Aleksej"https://zbmath.org/authors/?q=ai:turnsek.aleksejLet \(M\) and \(N\) be inner product spaces over \(F\in \{\mathbb{R}, \mathbb{C}\}\). The famous Wigner's theorem says that any mapping \(f:M \to N\) which preserves the transition probability, i.e.,
\[
\| \langle f(x), f(y)\rangle\| = \|\langle x, y\rangle\|, \qquad x,y\in M, \tag{W}
\]
must be phase-equivalent to a linear or a conjugate-linear isometry.
Now suppose that a mapping \(f : M \to N\) satisfies (W) approximately. More precisely,
assume that
\[
\|\langle(x), f(y)\rangle \| - \|\langle x, y\rangle\| \leq \phi(x,y), \qquad x,y\in M,
\]
where \(\phi: M \times M \to [0,\infty)\) is an appropriate control function.
The authors investigate the superstability of above inequality. In particular, they find that if \(f\) is surjective
all solutions of above inequality are in fact solutions of (W).A variant of Wigner's theorem in normed spaceshttps://zbmath.org/1472.390502021-11-25T18:46:10.358925Z"Ilišević, Dijana"https://zbmath.org/authors/?q=ai:ilisevic.dijana"Turnšek, Aleksej"https://zbmath.org/authors/?q=ai:turnsek.aleksejLet \(X\) and \(Y\) be normed spaces over \(\mathbb{F}\) and let \(U:X \rightarrow Y\) be a linear (or a conjugate linear) isometry. If a function \(f:X \rightarrow Y\) has the property \[ f(x)=\sigma(x) Ux, \quad x\in H, \] where \(\sigma\) is a phase function, i.e., \(\sigma\) takes values in modulus one scalars, then a function \(f\) is called phase equivalent to a linear (or a conjugate linear) isometry.
The main result of this paper is given in the following theorem.
Theorem. Let \(X\) and \(Y\) be normed spaces over \(\mathbb{F}\) and \(f:X \rightarrow Y\) a surjective mapping. Suppose that for all semi-inner products on \(X\) and \(Y\), we have \[ | [f(x),f(y)]| = |[x,y]|, \quad x,y \in X.\]
The following statements hold true:
\begin{itemize}
\item[(i)] If dim \(X=1\), then \(f\) is phase equivalent to a linear surjective isometry;
\item[(ii)] If dim \(X\geq 2\) and \(\mathbb{F} = \mathbb{R}\), then \(f\) is phase equivalent to a linear surjective isometry;
\item[(iii)] If dim \(X\geq 2\) and \(\mathbb{F} =\mathbb{C}\), then \(f\) is phase equivalent to a linear or conjugate linear surjective isometry.
\end{itemize}Korovkin-type approximation by operators in Riesz spaces via power series methodhttps://zbmath.org/1472.410102021-11-25T18:46:10.358925Z"Chil, Elmiloud"https://zbmath.org/authors/?q=ai:chil.elmiloud"Assili, Marwa"https://zbmath.org/authors/?q=ai:assili.marwaSummary: In this paper we prove an Ozguç, Yurdakadim and Taş version of the Korovkin-type approximation by operators in the sense of the power series method. That is, we try to extend the Korovkin approximation theorems, obtained by Ozguç and Taş in 2016, and Taş and Yurdakadim in 2017, for concrete classes of Banach spaces to the class of Riesz spaces. Some applications are presented.Monotone path-connectedness of Chebyshev sets in three-dimensional spaceshttps://zbmath.org/1472.410192021-11-25T18:46:10.358925Z"Alimov, Alexey R."https://zbmath.org/authors/?q=ai:alimov.alexey-r"Bednov, Borislav B."https://zbmath.org/authors/?q=ai:bednov.borislav-borisovichThe set of all nearest points (elements of best approximation) in a nonempty subset \(M\) of a normed linear space \(X\) to a given \(x\in X\) is denoted by \(P_Mx\). \(M\) is a Chebyshev set if \(P_Mx\) is a singleton for each \(x\in X\).
In this paper the authors show that in any three-dimensional space of the form \(Y\oplus _{\infty }\mathbb{R}\) (where dim \(Y = 2\)) an arbitrary Chebyshev set is monotone path-connected. Monotone path-connectedness is weaker than convexity and stronger than path-connectedness. Here, we write its definition, some others, and the statement of the main theorem of this paper, which, by the way, ends posing three problems.
Definition. A continuous curve \(k(\tau ), 0\leq \tau \leq 1\), in a normed linear space \(X\) is called monotone if the scalar function \(f(k(\tau ))\) is monotone in \(\tau \) for any extreme functional \(f\in \text{ext} S^*\) (\(\text{ext} S^*\) is the set of extreme points of the dual unit sphere \(S^*\)). A closed set is called monotone path-connected if any two points of this set can be connected by a continuous monotone curve lying in this set.
Definition. A point \(x\in X \backslash M\) is called a solar point if there exists a point \(y \in P_Mx\) (a luminosity point) such that \(y\in P_M( (1-\lambda y) +\lambda x)\) for all \(\lambda \geq 0\). A set \(M\subset X\) is a sun if any point \(x\in X \backslash M\) is a solar point for \(M\).
Remark 1. On a normed plane each sun is monotone path-connected.
Definition. A point \(s\) on the boundary of the unit ball \(B\) of the space \(X\) is called a smooth point of \(B\) (of the unit sphere \(S\)) if the support hyperplane to the ball \(B\) at the point \(s\) is unique. A point \(s\in S\) is an exposed point of \(B\) (of \(S\)) if there exists a support hyperplane \(H\) to the ball \(B\) at \(s\) such that \(H\cap B = \{ s\}\).
Theorem. In a three-dimensional normed space \(X\), each Chebyshev set is monotone path-connected if and only if one of the following two conditions holds:
\begin{enumerate}
\item each exposed point of the unit sphere of \(X\) is a smooth point;
\item \(X=Y\oplus _{\infty }\mathbb{R}\).
\end{enumerate}Example of divergence of a greedy algorithm with respect to an asymmetric dictionaryhttps://zbmath.org/1472.410202021-11-25T18:46:10.358925Z"Borodin, P. A."https://zbmath.org/authors/?q=ai:borodin.petr-aSummary: We construct an example of an asymmetric dictionary \(D\) in a Hilbert space \(H\) such that the linear combinations of elements of \(D\) with positive coefficients are dense in \(H\), but the greedy algorithm with respect to \(D\), in which the inner product with elements of \(D\) (not the modulus of this inner product) is maximized at each step, diverges for some initial element.Correction to: ``Some trigonometric polynomials with extremely small uniform norm and their applications''https://zbmath.org/1472.420012021-11-25T18:46:10.358925ZCorrection to the article [\textit{A. O. Radomskii}, Izv. Math. 84, No. 2, 361--391 (2020; Zbl 1440.42002); translation from Izv. Ross. Akad. Nauk, Ser. Mat. 84, No. 2, 166--196 (2020)].Characterization of Fourier transform of \(H\)-valued functions on the real linehttps://zbmath.org/1472.420052021-11-25T18:46:10.358925Z"Biswas, Md Hasan Ali"https://zbmath.org/authors/?q=ai:biswas.md-hasan-ali"Radha, Ramakrishnan"https://zbmath.org/authors/?q=ai:radha.ramakrishnanSummary: A characterization is obtained for the Fourier transform of functions belonging to \(\mathcal{\mathcal{L}}^2 ( \mathbb{R} , H )\), where \(H\) denotes a Hilbert \(C^\ast \)-module. But in the case of functions belonging to \(L^1 ( \mathbb{R} , H )\) a similar result is proved when \(H\) is a separable Hilbert space.Evaluation formula and approximation for Wiener integrals via the Fourier-type functionalhttps://zbmath.org/1472.420062021-11-25T18:46:10.358925Z"Chung, Hyun Soo"https://zbmath.org/authors/?q=ai:chung.hyun-soo"Lee, Un Gi"https://zbmath.org/authors/?q=ai:lee.un-giSummary: In order to calculate the Wiener integrals for functionals on Wiener space, one can usually apply the change of variable theorem. But, there are many functionals that are difficult or impossible to calculate even when using the change of variable formula. In order to solve this problem, we establish an evaluation formula via the Fourier-type functionals on Wiener space. We then present various examples to which our evaluation formula can be applied and with the corresponding numerical approximations.The boundedness of commutators of generalized fractional integral operators on specific generalized Morrey spaceshttps://zbmath.org/1472.420152021-11-25T18:46:10.358925Z"Budhi, Wono Setya"https://zbmath.org/authors/?q=ai:setya-budhi.wono|budhi.wono-setya"Lindiarni, Janny"https://zbmath.org/authors/?q=ai:lindiarni.jannySummary: In this note, we prove the boundedness of commutators of generalized fractional integral operators on the specific generalized Morrey spaces with different growth of functions. We call it specific Morrey spaces because the growth function relating with the kernel of integral operators.Boundedness of one-sided oscillatory integral operators on weighted Lebesgue spaceshttps://zbmath.org/1472.420172021-11-25T18:46:10.358925Z"Fu, Zunwei"https://zbmath.org/authors/?q=ai:fu.zunwei"Lu, Shanzhen"https://zbmath.org/authors/?q=ai:lu.shanzhen"Pan, Yibiao"https://zbmath.org/authors/?q=ai:pan.yibiao"Shi, Shaoguang"https://zbmath.org/authors/?q=ai:shi.shaoguangSummary: We consider one-sided weight classes of Muckenhoupt type, but larger than the classical Muckenhoupt classes, and study the boundedness of one-sided oscillatory integral operators on weighted Lebesgue spaces using interpolation of operators with change of measures.Multilinear commutators of Calderón-Zygmund operator on generalized weighted Morrey spaceshttps://zbmath.org/1472.420182021-11-25T18:46:10.358925Z"Guliyev, Vagif S."https://zbmath.org/authors/?q=ai:guliyev.vagif-sabir"Alizadeh, Farida Ch."https://zbmath.org/authors/?q=ai:alizadeh.farida-chSummary: The boundedness of multilinear commutators of Calderón-Zygmund operator \(T_{\vec{b}}\) on generalized weighted Morrey spaces \(M_{p,\varphi}(w)\) with the weight function \(w\) belonging to Muckenhoupt's class \(A_p\) is studied. When \(1<p<\infty\) and \(\vec{b}=(b_1, \dots, b_m)\), \(b_i \in \mathrm{BMO}\), \(i=1,\dots, m\), the sufficient conditions on the pair \((\varphi_1,\varphi_2)\) which ensure the boundedness of the operator \(T_{\vec{b}}\) from \(M_{p,\varphi_1}(w)\) to \(M_{p,\varphi_2}(w)\) are found. In all cases the conditions for the boundedness of \(T_{\vec{b}}\) are given in terms of Zygmund-type integral inequalities on \((\varphi_1,\varphi_2)\), which do not assume any assumption on monotonicity of \(\varphi_1(x,r)\), \(\varphi_2(x,r)\) in \(r\).On integral operators in weighted grand Lebesgue spaces of Banach-valued functionshttps://zbmath.org/1472.420212021-11-25T18:46:10.358925Z"Kokilashvili, Vakhtang"https://zbmath.org/authors/?q=ai:kokilashvili.vakhtang-m"Meskhi, Alexander"https://zbmath.org/authors/?q=ai:meskhi.alexanderSummary: The paper deals with boundedness problems of integral operators in weighted grand Bochner-Lebesgue spaces. We will treat both cases: when a weight function appears as a multiplier in the definition of the norm, or when it defines the absolute continuous measure of integration. Along with the diagonal case, we deal with the off-diagonal case. To get the appropriate result for the Hardy-Littlewood maximal operator, we rely on the reasonable bound of the sharp constant in the Buckley-type theorem, which is also derived in the paper.Boundedness of multilinear singular integrals on central Morrey spaces with variable exponentshttps://zbmath.org/1472.420242021-11-25T18:46:10.358925Z"Wang, Hongbin"https://zbmath.org/authors/?q=ai:wang.hongbin.1|wang.hongbin"Xu, Jingshi"https://zbmath.org/authors/?q=ai:xu.jingshi"Tan, Jian"https://zbmath.org/authors/?q=ai:tan.jian.2|tan.jian|tan.jian.1The authors prove boundedness for a class of multi-sublinear singular integral operators on the product of central Morrey spaces with variable exponents (see Theorem 2.1). As applications, for \(b=(b_1,\dots, b_m)\) such that every \(b_i\) is in the centeral BMO space with variable exponents, the authors further obtain boundedness of the multilinear Calderón-Zygmund commutator \([\mathfrak{b}, T]\) and \(T_{\mathfrak{b}}\) on the product of central Morrey spaces with variable exponents (see Theorems 3.1 and 3.2), where \[[\mathfrak{b}, T]f(x)= \int_{(\mathbb R^n)^m} K(x, y_1, \dots, y_m)\prod_{i=1}^m (b_i(x)-b_i(y_i)) f_i(y_i)\, dy_1\cdots\, dy_m\] and \[T_{\mathfrak{b}}f(x)=\int_{\mathbb R^n} \prod_{i=1}^m (b_i(x)-b_i(y))K(x,y)f(y)\, dy.\]Variation inequalities for rough singular integrals and their commutators on Morrey spaces and Besov spaceshttps://zbmath.org/1472.420262021-11-25T18:46:10.358925Z"Zhang, Xiao"https://zbmath.org/authors/?q=ai:zhang.xiao"Liu, Feng"https://zbmath.org/authors/?q=ai:liu.feng.4|liu.feng|liu.feng.3|liu.feng.2|liu.feng.5|liu.feng.1"Zhang, Huiyun"https://zbmath.org/authors/?q=ai:zhang.huiyunSummary: This paper is devoted to investigating the boundedness, continuity and compactness for variation operators of singular integrals and their commutators on Morrey spaces and Besov spaces. More precisely, we establish the boundedness for the variation operators of singular integrals with rough kernels \(\varOmega \in L^q (S^{n-1})\) \((q > 1)\) and their commutators on Morrey spaces as well as the compactness for the above commutators on Lebesgue spaces and Morrey spaces. In addition, we present a criterion on the boundedness and continuity for a class of variation operators of singular integrals and their commutators on Besov spaces. As applications, we obtain the boundedness and continuity for the variation operators of Hilbert transform, Hermit Riesz transform, Riesz transforms and rough singular integrals as well as their commutators on Besov spaces.On the regularity of the maximal function of a BV functionhttps://zbmath.org/1472.420312021-11-25T18:46:10.358925Z"Lahti, Panu"https://zbmath.org/authors/?q=ai:lahti.panuSummary: We show that the non-centered maximal function of a BV function is quasicontinuous. We also show that if the non-centered maximal functions of an SBV function is a BV function, then it is in fact a Sobolev function. Using a recent result of \textit{J. Weigt} [``Variation of the uncentered maximal characteristic function'', Preprint, \url{arXiv:2004.10485}], we are in particular able to show that the non-centered maximal function of a set of finite perimeter is a Sobolev function.The dual conjecture of Muckenhoupt and Wheedenhttps://zbmath.org/1472.420332021-11-25T18:46:10.358925Z"Osękowski, Adam"https://zbmath.org/authors/?q=ai:osekowski.adamSummary: Let \(T\) be a Calderón-Zygmund operator on \(\mathbb{R}^d\). We prove the existence of a constant \(C_{T,d} < \infty\) such that for any weight \(w\) on \(\mathbb{R}^d\) satisfying Muckenhoupt's condition \(A_1\), we have \[w\left(\{x\in \mathbb{R}^d:|Tf(x)| > w(x)\}\right) \leq C_{T,d}[w]_{A_1}\int_{\mathbb{R}^d}f \ \mathrm{d}x.\] The linear dependence on \([w]_{A_1} \), the \(A_1\) characteristic of \(w\), is optimal. The proof exploits the associated dimension-free inequalities for dyadic shifts.Endpoint regularity of the discrete multisublinear fractional maximal operatorshttps://zbmath.org/1472.420352021-11-25T18:46:10.358925Z"Zhang, Xiao"https://zbmath.org/authors/?q=ai:zhang.xiaoThe main results of this article are about the discrete centered and uncentered \(m\)-sublinear fractional maximal operators, \( \mathfrak{M}_\alpha \) and \( \widetilde{ \mathfrak{M} }_\alpha \) (respectively). First, a theorem on endpoint regularity is obtained for \( \widetilde{ \mathfrak{M} }_\alpha \) from the \(m\)-fold Cartesian product of \( \text{BV}(\mathbb{Z}) \) (\(0 \leq \alpha < 1 \)) or \( \ell^1 (\mathbb{Z}) \) (\( m-1 \leq \alpha < m \)) into the space of functions with bounded \(q\)-variation, where \(q\) depends on the subscript \(\alpha\). Here \( \text{BV}(\mathbb{Z}) \) denotes the space of functions of bounded variation defined on \(\mathbb{Z}\). After that, a theorem showing the boundedness of \(\mathfrak{M}_\alpha \) and \( \widetilde{ \mathfrak{M} }_\alpha \), \( 0 \leq \alpha < m \), from the \(m\)-fold Cartesian product of \( \ell^1 (\mathbb{Z}) \) into \( \text{BV}(\mathbb{Z}) \) is obtained.
Both theorems can be considered as the discrete version of the corresponding results contained in [\textit{F. Liu} and \textit{H. Wu}, Can. Math. Bull. 60, No. 3, 586--603 (2017; Zbl 1372.42015)] and also extend the corresponding already known results of \textit{E. Carneiro} and \textit{J. Madrid} [Trans. Am. Math. Soc. 369, No. 6, 4063--4092 (2017; Zbl 1370.26022)] and \textit{F. Liu} [Bull. Aust. Math. Soc. 95, No. 1, 108--120 (2017; Zbl 1364.42020)].On the Helmholtz decompositions of vector fields of bounded mean oscillation and in real Hardy spaces over the half spacehttps://zbmath.org/1472.420362021-11-25T18:46:10.358925Z"Giga, Yoshikazu"https://zbmath.org/authors/?q=ai:giga.yoshikazu"Gu, Zhongyang"https://zbmath.org/authors/?q=ai:gu.zhongyangSummary: This paper is concerned with the Helmholtz decompositions of vector fields of bounded mean oscillation over the half space and vector fields in real Hardy spaces over the half space. It proves the Helmholtz decomposition for vector fields of bounded mean oscillation over the half space whereas a partial Helmholtz decomposition for vector fields in real Hardy spaces over the half space. Meanwhile, it also establishes two sets of theories of real Hardy spaces over the half space which are compatible with the theory of \textit{A. Miyachi} [Stud. Math. 96, No. 3, 205--228 (1990; Zbl 0716.42017)].Duality for outer \(L^p_\mu (\ell^r)\) spaces and relation to tent spaceshttps://zbmath.org/1472.420372021-11-25T18:46:10.358925Z"Fraccaroli, Marco"https://zbmath.org/authors/?q=ai:fraccaroli.marcoIn this paper the author studies the outer \(L^p\) spaces introduced by \textit{Y. Do} and \textit{C. Thiele} [Bull. Am. Math. Soc., New Ser. 52, No. 2, 249--296 (2015; Zbl 1318.42016)] on sets endowed with a measure and an outer measure.
The author proves that in the case of finite sets, for \(1 < p \leq \infty\), \( 1 \leq r < \infty\) or \(p=r \in \{ 1,\ \infty\}\), the outer \(L_{\mu}^{p}(l^{r})\) quasi-norm is equivalent to a norm, and \(L_{\mu}^{p}(l^r)\) space is the Köthe dual space of \(L_{\mu}^{p^\prime}(l^{r^\prime})\). The author also shows that in the upper half space setting the above properties hold true in the full range \(1 \leq p\), \(r \leq \infty\) and establishes the equivalence between the classical tent space \(T^p_r\) and the outer \(L_{\mu}^{p}(l^r)\) space in the upper half space. Furthermore, the author gives a full classification of weak and strong type estimates for a class of embedding maps from classical \(L^p\) spaces in \(\mathbb{R}^n\) to outer \(L^{p}(l^r)\) spaces in the upper half space with a fractional scale factor.On the Riesz potential and its commutators on generalized Orlicz-Morrey spaceshttps://zbmath.org/1472.420382021-11-25T18:46:10.358925Z"Guliyev, Vagif S."https://zbmath.org/authors/?q=ai:guliyev.vagif-sabir"Deringoz, Fatih"https://zbmath.org/authors/?q=ai:deringoz.fatihSummary: We consider generalized Orlicz-Morrey spaces \(M_{\Phi,\varphi}(\mathbb R^n)\) including their weak versions \(WM_{\Phi,\varphi}(\mathbb R^n)\). In these spaces we prove the boundedness of the Riesz potential from \(M_{\Phi,\varphi_1}(\mathbb R^n)\) to \(M_{\Psi,\varphi_2}(\mathbb R^n)\) and from \(M_{\Phi,\varphi_1}(\mathbb R^n)\) to \(WM_{\Psi,\varphi_2}(\mathbb R^n)\). As applications of those results, the boundedness of the commutators of the Riesz potential on generalized Orlicz-Morrey space is also obtained. In all the cases the conditions for the boundedness are given either in terms of Zygmund-type integral inequalities on \((\varphi_1, \varphi_2)\), which do not assume any assumption on monotonicity of \(\varphi_1(x, r)\), \(\varphi_2(x, r)\) in \(r\).On the Schur-Horn problemhttps://zbmath.org/1472.420412021-11-25T18:46:10.358925Z"Abtahi, Fatemeh"https://zbmath.org/authors/?q=ai:abtahi.fatemeh"Kamali, Zeinab"https://zbmath.org/authors/?q=ai:kamali.zeinab"Keyshams, Zahra"https://zbmath.org/authors/?q=ai:keyshams.zahraSummary: Let \(\mathcal{H}\) be a separable Hilbert space. Recently, the concept of \(K\)-\(g\)-frame was introduced as a special generalization of \(g\)-Bessel sequences. In this paper, we point out some gaps in the proof of some existent results concerning \(K\)-\(g\)-frame. We present examples to indicate that these results are not necessarily valid. Then we remove the gaps and provide some desired conclusions. In this respect, we deal with Schur-Horn problem, which characterizes sequences \(\{\parallel f_n \parallel^2\}_{n = 1}^\infty \), for all frames \(\{f_n\}_{n = 1}^\infty\) with the same frame operator. We introduce the concept of synthesis related frames. Finally, as the main result, we investigate around Schur-Horn problem, for the case where \(\mathcal{H}\) is finite dimensional. In fact, we prove that two frames have the same frame operator if and only if they are synthesis related.Continuous Schauder frames for Banach spaceshttps://zbmath.org/1472.420442021-11-25T18:46:10.358925Z"Eisner, Joseph"https://zbmath.org/authors/?q=ai:eisner.joseph"Freeman, Daniel"https://zbmath.org/authors/?q=ai:freeman.daniel-h-junContinuous Schauder frames for a Banach space \(X\) were introduced in this paper. This concept generalizes the concept of continuous frames for Hilbert spaces as well as that of unconditional Schauder frames for Banach spaces. It was proved that some basic properties remain to be true in this general setting. Several equivalent conditions were obtained for shrinking properties, and/or boundedly completeness of continuous Schauder frames. In particular, it was proved that the reflexivity of the Banach space \(X\) is equivalent to the shrinking and boundedly completeness property of a continuous Schauder frame.Frame spectral pairs and exponential baseshttps://zbmath.org/1472.420452021-11-25T18:46:10.358925Z"Frederick, Christina"https://zbmath.org/authors/?q=ai:frederick.christina"Mayeli, Azita"https://zbmath.org/authors/?q=ai:mayeli.azitaSummary: Given a domain \(\varOmega \subset\mathbb{R}^d\) with positive and finite Lebesgue measure and a discrete set \(\varLambda \subset\mathbb{R}^d\), we say that \((\varOmega , \varLambda )\) is a frame spectral pair if the set of exponential functions \(\mathcal{E}(\varLambda ):=\{e^{2\pi i \lambda \cdot x}: \lambda \in \varLambda \}\) is a frame for \(L^2(\varOmega )\). Special cases of frames include Riesz bases and orthogonal bases. In the finite setting \(\mathbb{Z}_N^d\), \(d\), \(N\ge 1\), a frame spectral pair can be similarly defined. In this paper we show how to construct and obtain new classes of frame spectral pairs in \(\mathbb{R}^d\) by ``adding'' a frame spectral pair in \(\mathbb{R}^d\) to a frame spectral pair in \(\mathbb{Z}_N^d\). Our construction unifies the well-known examples of exponential frames for the union of cubes with equal volumes. We also remark on the link between the spectral property of a domain and sampling theory.Invariant subsets and homological properties of Orlicz modules over group algebrashttps://zbmath.org/1472.430022021-11-25T18:46:10.358925Z"Üster, Rüya"https://zbmath.org/authors/?q=ai:uster.ruya"Öztop, Serap"https://zbmath.org/authors/?q=ai:oztop.serapThe authors study some invariant subsets and homological properties of Orlicz modules over group algebras. In fact, for a locally compact group \(G\), a closed convex subset \(C\subseteq L^{\phi}(G)\) is left invariant if and only if \(h\ast C\subseteq C\) for all \(h\in P_{1}(G)\). The authors also show that for a locally compact group \(G\), each closed convex left invariant non-empty subset of \(L^{\phi}(G)\) contains the origin iff \(G\) is noncompact. As an interesting result in Banach homology, the authors show that \(L^{\phi}(G)\) is \(L^{1}(G) \)-projective iff \(G\) is compact.Approximate identity in closed codimension one ideals of semigroup algebrashttps://zbmath.org/1472.430032021-11-25T18:46:10.358925Z"Mohammadzadeh, B."https://zbmath.org/authors/?q=ai:mohammadzadeh.bahar|mohammadzadeh.bahramSummary: Let \(S\) be a foundation semigroup with identity and \(M_a(S)\) be its semigroup algebra. In this paper, we give necessary and sufficient conditions for the existence of a bounded approximate identity in closed codimension one ideals of semigroup algebra \(M_a(S)\) of a locally compact topological foundation semigroup with identity.Some remarks on weak Wiener-Ditkin setshttps://zbmath.org/1472.430062021-11-25T18:46:10.358925Z"Muraleedharan, T. K."https://zbmath.org/authors/?q=ai:muraleedharan.thettath-kSummary: Some properties of weak Wiener-Ditkin sets are studied and their local behavior and connections with the $C$-set-$S$-set problem are discussed.Lebedev-Skalskaya transforms on certain function spaces and associated pseudo-differential operatorshttps://zbmath.org/1472.440022021-11-25T18:46:10.358925Z"Mandal, U. K."https://zbmath.org/authors/?q=ai:mandal.upain-kumar"Prasad, Akhilesh"https://zbmath.org/authors/?q=ai:prasad.akhileshThe paper presents one of the index transforms known as the Lebedev-Skalskaya transform (LS-transform) given by \textit{S.~B. Yakubovich} in [Index transforms. London: World Scientific (1996; Zbl 0845.44001)] whose kernel is the modified Bessel function of the second kind. The authors of the present paper reflect the recent developments for the LS transform given by many researchers which can be seen in the exhaustive list of references.
This paper consists of five sections. Section~1 presents the introduction of the LS transform and its inversion. Two differential operators (see (1.22) and (1.23)) are introduced. The basic properties such as Plancherel's and Parseval's relation corresponding to the LS transform and adjoint LS transform are shown. Employing the translation operator, the convolution for the LS transform and other related properties are discussed. The operational formulas associated with the differential operator, translation and mainly convolution operator are represented in Section~2, and they are estimated in Lebesgue spaces for the LS transform. The continuity of the LS transform in the Lebesgue space is shown and then related to some other function spaces, dealt with in Section~3. The pseudodifferential operators in terms of the LS transform are defined in Section~4 and studied further for the integral representation of pseudodifferential operators.
Section~5 is devoted to an application. Using the LS transform, an integral equation with convolution kernel and an initial value problem are solved.Wave packet transform and fractional wave packet transform of rapidly decreasing functionshttps://zbmath.org/1472.440032021-11-25T18:46:10.358925Z"Thanga Rejini, M."https://zbmath.org/authors/?q=ai:thanga-rejini.m"Subash Moorthy, R."https://zbmath.org/authors/?q=ai:subash-moorthy.rErratum to: ``Nonseparable closed vector subspaces of separable topological vector spaces''https://zbmath.org/1472.460012021-11-25T18:46:10.358925Z"Kąkol, Jerzy"https://zbmath.org/authors/?q=ai:kakol.jerzy"Leiderman, Arkady G."https://zbmath.org/authors/?q=ai:leiderman.arkady-g"Morris, Sidney A."https://zbmath.org/authors/?q=ai:morris.sidney-aErratum to the authors' paper [ibid. 182, No. 1, 39--47 (2017; Zbl 1361.46001)].Double sequence spaces by means of Orlicz functionshttps://zbmath.org/1472.460022021-11-25T18:46:10.358925Z"Alotaibi, Abdullah"https://zbmath.org/authors/?q=ai:alotaibi.abdullah-m"Mursaleen, M."https://zbmath.org/authors/?q=ai:mursaleen.mohammad|mursaleen.momammad"Raj, Kuldip"https://zbmath.org/authors/?q=ai:raj.kuldipSummary: We define some classes of double entire and analytic sequences by means of Orlicz functions. We study some relevant algebraic and topological properties. Further some inclusion relations among the classes are also examined.Isomorphic universality and the number of pairwise nonisomorphic models in the class of Banach spaceshttps://zbmath.org/1472.460032021-11-25T18:46:10.358925Z"Džamonja, Mirna"https://zbmath.org/authors/?q=ai:dzamonja.mirnaSummary: We develop the framework of \textit{natural spaces} to study isomorphic embeddings of Banach spaces. We then use it to show that a sufficient failure of the generalized continuum hypothesis implies that the universality number of Banach spaces of a given density under a certain kind of positive embedding (\textit{very positive embedding}) is high. An example of a very positive embedding is a positive onto embedding between \(C(K)\) and \(C \left(L\right)\) for 0-dimensional \(K\) and \(L\) such that the following requirement holds for all \(h \neq 0\) and \(f \geq 0\) in \(C(K)\): if \(0 \leq T h \leq T f\), then there are constants \(a \neq 0\) and \(b\) with \(0 \leq a \cdot h + b \leq f\) and \(a \cdot h + b \neq 0\).A note on the stability of nonsurjective \(\varepsilon\)-isometries of Banach spaceshttps://zbmath.org/1472.460042021-11-25T18:46:10.358925Z"Cheng, Lixin"https://zbmath.org/authors/?q=ai:cheng.lixin"Dong, Yunbai"https://zbmath.org/authors/?q=ai:dong.yunbaiLet $X$ and $Y$ be Banach spaces and let $f: X \rightarrow Y$ be a standard $\varepsilon$-isometry for some $\varepsilon > 0$, i.e., $$ \left | \left \| f(x) - f(y) \right \| - \left \| x - y \right \| \right | \leq \varepsilon \quad \text{for all }x,y \in X$$ and $f(0) =0$. A standard $\varepsilon$-isometry $f: X \rightarrow Y$ is said to be stable if there exist a linear operator $T: \overline{\text{span}}(f(X)) \rightarrow X$ and a constant $\gamma > 0$ such that $$ \left \| Tf(x) -x\right \| \leq \gamma \varepsilon \quad \text{for all $x \in X$.}$$ \textit{S.-W. Qian} [Proc. Am. Math. Soc. 123, 1797--1803 (1995; Zbl 0827.47022)] studied the stability of $\varepsilon$-isometries and showed that if $X$ is a closed uncomplemented subspace of $Y$, then there is an unstable standard $\varepsilon$-isometry $f = \mathrm{Id}_X + \varepsilon g$, where $g: X \rightarrow B_Y$ is a (not necessarily continuous) bijection. This implies that it is difficult to claim the stability property of an $\varepsilon$-isometry defined on general Banach spaces.
\textit{L.-X. Cheng} et al. [J. Funct. Anal. 264, 713--734 (2013; Zbl 1266.46008)] showed a weak stability theorem for non-surjective $\varepsilon$-isometries: Suppose that $f:X \rightarrow Y$ is a standard $\varepsilon$-isometry. Then for every $x^* \in X^*$ there exists $\varphi \in Y^*$ with $\|\varphi\| = \| x^*\| =: r$ such that $$ | \langle x^*,x\rangle - \langle \varphi, f(x)\rangle | \leq 4 r \varepsilon \quad \text{for all }x \in X.$$ \par The authors show that the constant ``4'' in the aforementioned theorem can be improved to ``3'' and give an example that shows that this is the best estimate in general Banach spaces.Some remarks on the weak maximizing propertyhttps://zbmath.org/1472.460052021-11-25T18:46:10.358925Z"Dantas, Sheldon"https://zbmath.org/authors/?q=ai:dantas.sheldon"Jung, Mingu"https://zbmath.org/authors/?q=ai:jung.mingu"Martínez-Cervantes, Gonzalo"https://zbmath.org/authors/?q=ai:martinez-cervantes.gonzaloA pair \((E,F)\) of Banach spaces is said to have the {\em weak maximizing property} (wmp) if for every bounded linear operator \(T:E \to F\), \(T\) attains its norm whenever there is a non-weakly null sequence \((x_n)\) of norm one vectors such that \(\| T(x_n)\| \to \|T\|\). The idea for this property comes from a paper by \textit{D.~Pellegrino} and \textit{E.~V. Teixeira} [Bull. Braz. Math. Soc. (N.S.) 40, No.~3, 417--431 (2009; Zbl 1205.47012)], and it was recently adapted by \textit{R.~M. Aron} et al. [Proc. Am. Math. Soc. 148, No.~2, 741--750 (2020; Zbl 1442.46007)]. The latter paper contained several open questions, a number of which are addressed here.
For instance, although it was known that the pair \((\ell_p,\ell_q)\) has the wmp (\(1 < p < \infty$, $1 \leq q < \infty\)), it was unknown whether a similar property holds for \((L_p[0,1],L_q[0,1])\). The authors show that, in fact, wmp fails for \(p > 2\) or \(q < 2\). Among others, this result is a consequence of the authors' main result.
Theorem. Let \(E, F, X\), and \(Y\) be non-trivial Banach spaces such that not every bounded linear operator \(E \to F\) attains its norm. Then (i) whenever \(1 \leq q < p < \infty\), the pair \((E \oplus_p X, F \oplus_q Y)\) fails the wmp, and (ii) \((E \oplus_\infty X, F)\) fails the wmp. On the other hand, \((\ell_s \oplus_p \ell_p, \ell_s \oplus_q \ell_q)\) has the wmp if and only if \(1 < p \leq s \leq q < \infty\).
A number of interesting questions remain, such as: If \(E\) is a reflexive Banach space such that \((E,F)\) has the wmp for every \(F\), does it follow that \(E\) is finite dimensional? Also, the following is known: if the pair \((E, F)\) has the wmp, then an operator \(T +K\) attains its norm whenever a bounded linear operator \(T:E \to F\) and a compact operator \(K:E \to F\) satisfy \(\|T\| < \|T + K\|\). On the other hand, the authors note that the following converse is open: Let \(E\) and \(F\) be reflexive. Suppose that whenever \(T:E \to F\) (resp., \(K:E \to F\)) is a bounded (resp., compact) linear operator such that if \(\|T\| < \|T + K\|\), then necessarily \(T + K\) attains its norm. Does it follow that the pair \((E,F)\) has the wmp?Tingley's problem for \(p\)-Schatten von Neumann classeshttps://zbmath.org/1472.460062021-11-25T18:46:10.358925Z"Fernández-Polo, Francisco J."https://zbmath.org/authors/?q=ai:fernandez-polo.francisco-j"Jordá, Enrique"https://zbmath.org/authors/?q=ai:jorda.enrique"Peralta, Antonio M."https://zbmath.org/authors/?q=ai:peralta.antonio-mSummary: Let \(H\) and \(H'\) be the complex Hilbert spaces. For \(p\in (1,\infty)\setminus \{2\}\), we consider the Banach space \(C_p(H)\) of all \(p\)-Schatten-von Neumann operators, whose unit sphere is denoted by \(S(C_p(H))\). We prove that every surjective isometry \(\Delta:S(C_p(H))\to S(C_p(H'))\) can be extended to a complex linear or to a conjugate linear surjective isometry \(T:C_p(H)\to C_p(H')\).On the compact operators case of the Bishop-Phelps-Bollobás property for numerical radiushttps://zbmath.org/1472.460072021-11-25T18:46:10.358925Z"García, Domingo"https://zbmath.org/authors/?q=ai:garcia.domingo"Maestre, Manuel"https://zbmath.org/authors/?q=ai:maestre.manuel"Martín, Miguel"https://zbmath.org/authors/?q=ai:martin-suarez.miguel"Roldán, Óscar"https://zbmath.org/authors/?q=ai:roldan.oscar.1This work is devoted to study the Bishop-Phelps-Bollobás property for the numerical radius in the case of compact operators (BPBp-nu for compact operators in short). This research lies within the rapidly developing branch of Banach space theory which studies vector-valued versions of the Bishop-Phelps-Bollobás theorem. The interested reader will find a complete account of the state of the art on the matter in the detailed introduction of the paper under review.
The main result of the paper proves that \(C_0(L)\) spaces have the BPBp-nu for compact operators for every Hausdorff topological locally compact space \(L\), improving a previously known result for some real \(C(K)\) spaces. In the words of the authors, this is achieved by combining two main ingredients. The first one is an abstract way to lift the BPBp-nu for compact operators from a subspace to the whole space under some restrictions. As an elaborated corollary of this result, the authors obtain that isometric preduals of \(\ell_1\) have the BPBp-nu for compact operators. The second ingredient is some kind of strong approximation property of \(C_0(L)\) spaces and their duals, which is very interesting in itself. In particular, it allows to construct a net of norm-one projections on \(C_0(L)\), with ranges isometrically isomorphic to finite-dimensional \(\ell_\infty\) spaces, converging in the strong operator topology to the identity of \(C_0(L)\) and such that the net consisting of the adjoints of the projections converges in the strong operator topology to the identity of \(C_0(L)^*\). Finally, it is also shown that real Hilbert spaces have the BPBp-nu for compact operators.Approximate local isometries on spaces of absolutely continuous functionshttps://zbmath.org/1472.460082021-11-25T18:46:10.358925Z"Hosseini, Maliheh"https://zbmath.org/authors/?q=ai:hosseini.maliheh"Jiménez-Vargas, A."https://zbmath.org/authors/?q=ai:jimenez-vargas.antonioLet \(X,Y\) be compact sets of real numbers. Let \(AC(X)$ $(AC(Y))\) denote space of complex absolutely continous functions, equipped with the norm \(\|f\|_{\Sigma}= \|f\|_{\infty}+V(f)\), for \(f \in AC(X)\), where \(V(f)\) denotes the variation of~\(f\). A description of the surjective isometries between such spaces has long been known (see [\textit{V. D. Pathak}, Can. J. Math. 34, 298--306 (1982; Zbl 0464.46029)]). The authors first note that the group of isometries is not topologically reflexive and go on to provide a description of the objects in the topological closure of the isometry group (Theorem~2).Invariant means on Abelian groups capture complementability of Banach spaces in their second dualshttps://zbmath.org/1472.460092021-11-25T18:46:10.358925Z"Goucher, Adam P."https://zbmath.org/authors/?q=ai:goucher.adam-p"Kania, Tomasz"https://zbmath.org/authors/?q=ai:kania.tomaszThe following beautiful theorem is proven. Let \(X\) be a Banach space and \(\lambda\geq 1\). Then the following are equivalent:
\begin{itemize}
\item [(i)] \(X\) is \(\lambda\)-complemented in \(X^{\ast\ast}\);
\item [(ii)] for every amenable semigroup \(S\) there exists an \(X\)-valued invariant \(\lambda\)-mean on \(S\);
\item [(iii)] same as in (ii), but for every free abelian group of rank the cardinality of \(X^{\ast\ast}\);
\item [(iv)] the \(X\)-valued invariant \(\lambda\)-mean on \(S\), as in (ii) and (iii), exists on one particular group, namely, the additive group of \(X^{\ast\ast}\).
\end{itemize}
The implication (i) \(\Rightarrow\) (ii) is known, while the next two should be clear. That (iv) implies (iii) is in fact a rather quick observation, so the main content of the paper is to prove that (iii) implies (i).
Section 2 of the paper contains some auxiliary lemmas. In Section 3, the following interesting theorem is established: Let \(V\) be an infinite-dimensional vector space over an arbitrary field and let \(F(V)\) denote the set of finite-dimensional subspaces of \(V\). Then there exists a binary operation \(\ast\) on \(F(V)\) such that (i) (\(F(V),{\ast}\)) is a free commutative monoid, and (ii) for any \(F,G\in F(V)\), we have \(F,G\subseteq F\ast G\). Concequently, the Grothendieck group of (\(F(V),{\ast}\)) is a free abelian group.
This purely algebraic result in tandem with the principle of local reflexivity produces the sought for linear projection on \(X^{\ast\ast}\).A note on point-finite coverings by ballshttps://zbmath.org/1472.460102021-11-25T18:46:10.358925Z"De Bernardi, Carlo Alberto"https://zbmath.org/authors/?q=ai:de-bernardi.carlo-alberto\textit{V. P. Fonf} and \textit{C.~Zanco} proved the following theorem [Can. Math. Bull. 57, No.~1, 42--50 (2014; Zbl 1298.46015)].
{Theorem.} An infinite-dimensional separable Hilbert space does not admit a point-finite covering by closed non-degenerate balls.
Here, \textit{point-finite} means that every point of the Hilbert space is contained in finitely many elements of the covering. This paper greatly simplifies the proof of Fonf and Zanco's result and slightly generalizes it, namely, it also adds the following statement:
Moreover, if the density character of the Hilbert space is \(<2^\omega\), then it does not admit a point-finite covering consisting of (non-degenerate) open or closed balls.
The heart of the proof is the following theorem which is obtained through an adaptation of an argument by \textit{J. Lindenstrauss} and \textit{R. R. Phelps} [Isr. J. Math. 6, 39--48 (1968; Zbl 0157.43802)].
{Theorem.} Let \(X\) be an infinite-dimensional reflexive Banach space and \((f_n)\subset X^*\setminus U_{X^*}\) (where \(U_{X^*}\) denotes the open unit ball). For every \(n\), suppose \(S_n\) is either \[\{x\in X: f_n(x)\ge1\}\quad \text{ or }\quad \{x\in X:f_n(x)>1\}.\] If \(\{S_n:n\in\mathbb N\}\) covers the unit sphere of \(X\), then the covering is not point-finite.
In the second part of the paper, the main result is extended, using an argument by \textit{V.~P. Fonf} et al. [J. Geom. Anal. 24, No.~4, 1891--1897 (2014; Zbl 1315.46019)], to Banach spaces (not necessarily separable) that are both uniformly rotund and uniformly smooth.An attempt to measure diametrality of convex setshttps://zbmath.org/1472.460112021-11-25T18:46:10.358925Z"Goebel, Kazimierz"https://zbmath.org/authors/?q=ai:goebel.kazimierz"Prus, Stanisław"https://zbmath.org/authors/?q=ai:prus.stanislawSummary: Bounded, convex sets consisting only of diametral points, \textit{diametral sets}, exist only in some sense irregular Banach spaces. The spaces lacking such sets are said to have \textit{normal structure}. We propose here an attempt to measure how far are spaces or sets from \textit{diametrality}, meaning to contain such subsets. Basing on examples, we discuss properties, irregularities and open geometrical problems appearing in this approach.Diametrically complete sets with empty interior in reflexive Banach spaces with the nonstrict Opial propertyhttps://zbmath.org/1472.460122021-11-25T18:46:10.358925Z"Kaczor, Wiesława"https://zbmath.org/authors/?q=ai:kaczor.wieslawa-j"Kuczumow, Tadeusz"https://zbmath.org/authors/?q=ai:kuczumow.tadeusz"Reich, Simeon"https://zbmath.org/authors/?q=ai:reich.simeonSummary: We prove that for each reflexive Banach space \((X,\|\cdot\|_X)\) with the nonstrict Opial property, there exists an equivalent norm \(\|\cdot\|_1\) such that the Banach space \((X,\|\cdot\|_1)\) contains a diametrically complete set with empty interior.Corrigendum to: ``Angular equivalence of normed spaces''https://zbmath.org/1472.460132021-11-25T18:46:10.358925Z"Kikianty, Eder"https://zbmath.org/authors/?q=ai:kikianty.eder"Sinnamon, Gord"https://zbmath.org/authors/?q=ai:sinnamon.gordSummary: A correct proof is given for Theorem 2.1 of [the authors, ibid. 454, No. 2, 942--960 (2017; Zbl 1390.46013)].Some open problems related to fixed point properties and amenabilityhttps://zbmath.org/1472.460142021-11-25T18:46:10.358925Z"Lau, Anthony To-Ming"https://zbmath.org/authors/?q=ai:lau.anthony-to-mingSummary: In this paper, we post some open problems related to fixed point theory and amenability of semigroups mainly based on my joint work with Professor Wataru Takahashi.Operators reversing $b$-Birkhoff orthogonality in 2-normed linear spaceshttps://zbmath.org/1472.460152021-11-25T18:46:10.358925Z"Pirali, R."https://zbmath.org/authors/?q=ai:pirali.r"Momeni, M."https://zbmath.org/authors/?q=ai:momeni.mojgan|momeni.mostafa|momeni.maryam|momeni.mohsenSummary: In this paper, we discuss the relationships between 2-functionals and existence of $b$-Birkhoff orthogonal elements in 2-normed linear spaces. Moreover, we obtain some characterizations of 2-inner product spaces by $b$-Birkhoff orthogonality. Then we study the operators reversing $b$-Birkhoff orthogonality in 2-normed linear spaces.The Wigner property for CL-spaces and finite-dimensional polyhedral Banach spaceshttps://zbmath.org/1472.460162021-11-25T18:46:10.358925Z"Tan, Dongni"https://zbmath.org/authors/?q=ai:tan.dongni"Huang, Xujian"https://zbmath.org/authors/?q=ai:huang.xujianLet \(X\) and \(Y\) be real Banach spaces. A map \(f : X \to Y\) is a \textit{phase-isometry} if it satisfies, for all \(x,y \in X\),
\[
\{ \|f(x) + f(y)\|, \|f(x) - f(y)\| \} = \{ \|x + y\|, \|x - y\| \}.
\]
Two maps \(f,g : X \to Y\) are \textit{phase equivalent} if there exists a phase function \(\varepsilon: X \to \{-1,1\}\) such that \(g = \varepsilon \cdot f\).
A Banach space \(X\) has the \textit{Wigner property} if, for any Banach space \(Y\), every surjective phase-isometry \(f: X \to Y\) is phase-equivalent to a linear isometry.
The main result in the paper under review is that finite-dimensional polyhedral spaces and CL-spaces have the Wigner property. Recall that a (real) Banach space \(X\) is a CL-space if the unit ball \(B_X\) equals the convex hull of \(M \cup -M\) for every maximal convex subset \(M\) of the unit sphere \(S_X\). It is well known that all \(C(K)\)-spaces and \(L_1(\mu)\)-spaces are (real) CL-spaces. Note that the situation is different when considering Banach spaces over the complex numbers, see, e.g., \textit{M.~Martín} and \textit{R.~Payá} [Ark. Mat. 42, No.~1, 107--118 (2004; Zbl 1057.46020)].
The main tool used in this paper is the notion of a star-point that the authors introduce. Given \(x \in S_X\), the \textit{star of \(x\)} with respect to \(S_X\) is the set
\[
St(x) = \{y \in S_X : \|y + x\| = 2\},
\]
that is, all \(y \in S_X\) such that the line segment between \(x\) and \(y\) is contained in the sphere. A~point \(x \in S_X\) is a \textit{star point} of \(S_X\) if the star of \(x\), \(St(x)\), is a maximal convex subset of \(S_X\). It is shown that phase-isometries carries star points to star points.On relative \(k\)-uniform rotundity, normal structure and fixed point property for nonexpansive mapshttps://zbmath.org/1472.460172021-11-25T18:46:10.358925Z"Veena Sangeetha, M."https://zbmath.org/authors/?q=ai:veena-sangeetha.mSummary: The idea of \(k\)-uniform rotundity relative to a \(k\)-dimensional subspace generalizes the classical notion of uniform rotundity in a direction. A normed space that is \(k\)-uniformly rotund relative to every \(k\)-dimensional subspace is said to be URE\(_k\). In this article, relative \(k\)-uniform rotundity is used to obtain: (1) new conditions sufficient for asymptotic centers to be compact; (2) new equivalent conditions for normal structure, weak normal structure and weak fixed point property for nonexpansive maps (WFPP) in a normed space. These results are then applied to study the inheritance of some geometric and fixed point properties in products of normed spaces. In particular, it is proved that if \(N_1\), \(N_2\) are normed spaces such that \(N_1\) is URE\(_{k_1}\) and \(N_2\) is URE\(_{k_2}\), then for \(1<p<\infty\), the \(p\)-direct such of \(N_1\) and \(N_2\) is URE\(_{(k_1+k_2-1)}\). Also, it is proved that if \(N_1\) has WFPP and \(N_2\) is URE\(_k\) for some positive integer \(k\), then with respect to certain norms (including the standard \(p\)-norms for \(1<p<\infty\)) \(N_1\otimes N_2\) has WFPP. In addition to these, relative \(k\)-uniform rotundity in a class of subspaces of the Banach space of all bounded real valued functions on a nonempty set with supremum norm is also studied.On Lipschitz-free spaces over spheres of Banach spaceshttps://zbmath.org/1472.460182021-11-25T18:46:10.358925Z"Candido, Leandro"https://zbmath.org/authors/?q=ai:candido.leandro"Kaufmann, Pedro L."https://zbmath.org/authors/?q=ai:kaufmann.pedro-levit\textit{F. Albiac} et al. [Isr. J. Math. 240, No. 1, 65--98 (2020; Zbl 1462.46003)] have shown that for any metric space there is a bounded one such that the corresponding Lipschitz-free spaces are isomorphic. In case the metric space is a Banach space, one can be more specific. Indeed, \textit{P. L. Kaufmann} [Stud. Math. 226, No. 3, 213--227 (2015; Zbl 1344.46008)] proved that the Lipschitz-free space \(\mathcal F(X)\) is isomorphic to \(\mathcal F(B_X)\) for any Banach space \(X\). In this short paper, it is shown that the Lipschitz-free space on a Banach space \(X\) is isomorphic to the one on its sphere, provided that \(X\) is isomorphic to its hyperplanes.A remark on the interpolation inequality between Sobolev spaces and Morrey spaceshttps://zbmath.org/1472.460192021-11-25T18:46:10.358925Z"Tran, Minh-Phuong"https://zbmath.org/authors/?q=ai:tran.minh-phuong"Nguyen, Thanh-Nhan"https://zbmath.org/authors/?q=ai:nguyen.thanh-nhanSummary: Interpolation inequalities play an important role in the study of PDEs and their applications. There are still some interesting open questions and problems related to integral estimates and regularity of solutions to elliptic and/or parabolic equations. The main purpose of our work is to provide an important observation concerning the \(L^p\)-boundedness property in the context of interpolation inequalities between Sobolev and Morrey spaces, which may be useful for those working in this domain. We also construct a nontrivial counterexample, which shows that the range of admissible values of \(p\) is optimal in a certain sense. Our proofs rely on integral representations and on the theory of maximal and sharp maximal functions.Nonlinear weakly sequentially continuous embeddings between Banach spaceshttps://zbmath.org/1472.460202021-11-25T18:46:10.358925Z"Braga, B. M."https://zbmath.org/authors/?q=ai:de-mendonca-braga.brunoThe fundamental question of nonlinear geometry of Banach spaces is to determine to what extent the metric structure of a Banach space determines the linear structure of the space. One aspect of this is to study various types of nonlinear embeddings of one Banach space into another.
Let \((M_1,d_1)\) and \((M_2,d_2)\) be metric spaces and suppose \(f : M_1 \to M_2\) is any mapping. The modulus of continuity of \(f\) is defined, for \(t \ge 0\), by
\[
\omega_f(t) = \sup \{ d_2(f(x), f(y)) : d_1(x,y) \le t \},
\]
while the modulus of expansion of \(f\) is defined by
\[
\rho_f(t) = \inf \{ d_2(f(x), f(y)) : d_1(x,y) \ge t \}.
\]
The map \(f\) is said to be a coarse map if \(\omega_f(t) < \infty\) for all \(t \ge 0\), and \(f\) is a coarse embedding if it is coarse and \(\lim_{t \to \infty} \rho_f(t) =\infty\). Further, \(f\) is a uniform embedding if \(\lim_{t \to 0^+} \omega_f(t) = 0\) and \(\rho_f(t) > 0\) for all \(t > 0\).
The focus of this paper is the following weaker version of coarse and uniform embeddings: For Banach spaces \(X\) and \(Y\), consider weakly sequentially continuous maps \(f : X \to Y\) that are coarse and satisfy the following separation property:
\[
\exists \beta > \alpha > 0: \inf_{\|x-y\| \in [\alpha,\beta]}\|f(x)-f(y)\| > 0.
\]
It is shown that if \(1 \le p < q\), then there exists no weakly sequentially continuous map \(\ell_q \to \ell_p\) that is coarse and satisfies the above separation property. (But note that \(\ell_q\) both coarsely (and uniformly) embeds into \(\ell_p\) for all \(p,q \in [1,2]\) as shown by \textit{P.~W. Nowak} [Fundam. Math. 189, No.~2, 111--116 (2006; Zbl 1097.46052)].) It is also shown that if \(1 \le p < q\), then there exists an embedding \(\ell_p \to \ell_q\) which is both a coarse and uniform embedding (a strong embedding) and also weakly sequentially continuous.
The main result, Theorem~1.3, says that if a Banach space \(X\) can be mapped into a Banach space \(Y\) by a weakly sequentially continuous map that is coarse and satisfies the above separation property, then there exists \(c > 0\) so that if \((e_n)_n\) is a spreading model in a Banach space \((S,\|\cdot\|_S)\) of a normalized weakly null sequence in \(X\), then
\[
c \|e_1 + e_2 + \cdots + e_k\|_{\bar{\delta}_Y} \le \|e_1 + e_2 + \cdots + e_k\|_S
\]
for all \(k \in \mathbb{N}\). Here \(\bar{\delta}_Y\) is the modulus of asymptotic uniform convexity. This modulus \(\bar{\delta}_Y\) is equivalent to a convex function and the norm is the norm in the associated Orlicz space. \textit{N. J. Kalton} [Trans. Am. Math. Soc. 365, No. 2, 1051--1079 (2013; Zbl 1275.46011)] obtained the same conclusion as in Theorem~1.3 under the assumption that \(X\) coarse Lipschitz embeds into \(Y\).
As corollaries to the main theorem it is shown that if \(X\) maps into \(Y\) by a weakly sequentially continuous map that is coarse and satisfies the above separation property, then if \(Y\) is \(p\)-asymptotically uniformly convex for some \(p \in [1,\infty)\), then \(X\) must have the \(p\)-co-Banach-Saks property.
Let us also mention that it is shown that if \(X\) has a normalized weakly null sequence that generates a spreading model which is isomorphic to the standard basis of \(c_0\) and there exists a weakly sequentially continuous map \(X \to Y\) that is coarse and satisfies the above separation property, then \(Y\) does not have an equivalent asymptotically uniformly convex norm.
The paper ends with a list of open problems. Note that Problem~5.4 has been given a negative answer by \textit{B.~M. Braga} [J. Inst. Math. Jussieu 20, No.~1, 65--102 (2021; Zbl 1466.46014)].Order topology on orthocomplemented posets of linear subspaces of a pre-Hilbert spacehttps://zbmath.org/1472.460212021-11-25T18:46:10.358925Z"Buhagiar, D."https://zbmath.org/authors/?q=ai:buhagiar.david"Chetcuti, E."https://zbmath.org/authors/?q=ai:chetcuti.emmanuel"Weber, H."https://zbmath.org/authors/?q=ai:weber.hansSummary: Motivated by the Hilbert-space model for quantum mechanics, we define a pre-Hilbert space logic to be a pair \((S,\mathcal{L})\), where \(S\) is a pre-Hilbert space and \(\mathcal{L}\) is an orthocomplemented poset of orthogonally closed linear subspaces of \(S\), closed w.r.t. finite-dimensional perturbations (i.e., if \(M\in\mathcal{L}\) and \(F\) is a finite-dimensional linear subspace of \(S\), then \(M+F\in\mathcal{L})\). We study the order topology \(\tau_o(\mathcal{L})\) on \(\mathcal{L}\) and show that completeness of \(S\) can by characterized by the separation properties of the topological space \((\mathcal{L},\tau_o(\mathcal{L}))\). It will be seen that the remarkable lack of a proper probability theory on pre-Hilbert space logics -- for an incomplete \(S\) -- comes out elementarily from this topological characterization.Sequence space representations for spaces of entire functions with rapid decay on stripshttps://zbmath.org/1472.460222021-11-25T18:46:10.358925Z"Debrouwere, Andreas"https://zbmath.org/authors/?q=ai:debrouwere.andreasLet \(\omega:[0, \infty)\to [0, \infty]\) be an increasing continuous function with \(\omega(0) = 0\) and \(\log t = o(\omega(t))\) as \(t\) goes to \(\infty.\) The author considers the Fréchet space \({\mathcal U}_\omega({\mathbb C})\) consisting of those entire functions \(\varphi\) such that \[ \sup_{|{\Im z}| < n}|\varphi(z)| e^{n\omega\left(|{\Re z}|\right)} < \infty \ \quad \text{for all }n\in {\mathbb N}.\] The main results of the paper can be summarized as follows. \par
\begin{itemize}
\item[(a)] If \(\omega\) satisfies \(\omega(t+1) = O(\omega(t))\), then \({\mathcal U}_\omega({\mathbb C})\) is isomorphic to some power series space of infinite type.
\item[(b)] If \(\omega\) satisfies \(\omega(2t) = O(\omega(t))\), then \({\mathcal U}_\omega({\mathbb C})\) is isomorphic to \(\Lambda_\infty\left(\omega^\ast(n)\right),\) where \(\omega^\ast(n) = \left(n\omega^{-1}(n)\right)^{-1}\).
\end{itemize}
\par To prove (a), the author shows that \({\mathcal U}_\omega({\mathbb C})\) has properties (DN) and (\(\Omega\)). To conclude (b), the diametral dimension of \({\mathcal U}_\omega({\mathbb C})\) is calculated. This is done by combining a result due to \textit{M. Langenbruch} [Stud. Math. 233, No. 1, 85--100 (2016; Zbl 1356.46007)] with properties of the short-time Fourier transform. As a consequence, a sequence space representation for projective Gelfand-Shilov spaces is obtained.Weighted conformal invariance of Banach spaces of analytic functionshttps://zbmath.org/1472.460232021-11-25T18:46:10.358925Z"Aleman, Alexandru"https://zbmath.org/authors/?q=ai:aleman.alexandru"Mas, Alejandro"https://zbmath.org/authors/?q=ai:mas.alejandroSummary: We consider Banach spaces of analytic functions in the unit disc which satisfy a weighted conformal invariance property, that is, for a fixed \(\alpha > 0\) and every conformal automorphism \(\varphi\) of the disc, \(f \mapsto f \circ \varphi ( \varphi^\prime )^\alpha\) defines a bounded linear operator on the space in question, and the family of all such operators is uniformly bounded in operator norm. Many common examples of Banach spaces of analytic functions like Korenblum growth classes, Hardy spaces, standard weighted Bergman and certain Besov spaces satisfy this condition. The aim of the paper is to develop a general approach to the study of such spaces based on this property alone. We consider polynomial approximation, duality and complex interpolation, we identify the largest and the smallest as well as the ``unique'' Hilbert space satisfying this property for a given \(\alpha > 0\). We investigate the weighted conformal invariance of the space of derivatives, or anti-derivatives with the induced norm, and arrive at the surprising conclusion that they depend entirely on the properties of the (modified) Cesàro operator acting on the original space. Finally, we prove that this last result implies a John-Nirenberg type estimate for analytic functions \(g\) with the property that the integration operator \(f \mapsto \int_0^z f(t) g^\prime(t) d t\) is bounded on a Banach space satisfying the weighted conformal invariance property.On the geometry of Banach spaces of the form \(\mathrm{Lip}_0(C(K))\)https://zbmath.org/1472.460242021-11-25T18:46:10.358925Z"Candido, Leandro"https://zbmath.org/authors/?q=ai:candido.leandro"Kaufmann, Pedro L."https://zbmath.org/authors/?q=ai:kaufmann.pedro-levitA result of \textit{Y. Dutrieux} and \textit{V. Ferenczi} [Proc. Am. Math. Soc. 134, No. 4, 1039--1044 (2006; Zbl 1097.46004)] states that for any infinite metric compact space \(K\), the spaces \(\mathcal{F}(C(K))\) and \(\mathcal{F}(c_0)\) are isomorphic, although \(C(K)\) and \(c_0\) are not even uniformly homeomorphic in general, where \(\mathcal{F}(X)\) denotes the free-Lipschitz space associate to a pointed metric space \(X\). In this paper, sufficient conditions are established for a space \(\mathcal{F}(C(K))\) (for general compacta \(K\)) to be isomorphic to \(\mathcal{F}(c_0(\Gamma))\), for some uncountable \(\Gamma\).Atomic decomposition and Carleson measures for weighted mixed norm spaceshttps://zbmath.org/1472.460252021-11-25T18:46:10.358925Z"Peláez, José Ángel"https://zbmath.org/authors/?q=ai:pelaez.jose-angel"Rättyä, Jouni"https://zbmath.org/authors/?q=ai:rattya.jouni"Sierra, Kian"https://zbmath.org/authors/?q=ai:sierra.kianSummary: The purpose of this paper is to establish an atomic decomposition for functions in the weighted mixed norm space \(A^{p,q}_\omega\) induced by a radial weight \(\omega\) in the unit disc admitting a two-sided doubling condition. The obtained decomposition is further applied to characterize Carleson measures for \(A^{p,q}_\omega\), and bounded differentiation operators \(D^{(n)}(f)=f^{(n)}\) acting from \(A^{p,q}_\omega\) to \(L^s_\mu\), induced by a positive Borel measure \(\mu\), on the full range of parameters \(0<p,q,s<\infty\).Erratum to: ``On an inequality in Lebesgue space with mixed norm and with variable summability exponent''https://zbmath.org/1472.460262021-11-25T18:46:10.358925Z"Aliev, Rashid A."https://zbmath.org/authors/?q=ai:aliev.rashid-avazagaFrom the text: In [\textit{R. A. Bandaliev}, ibid. 84, No. 3, 303--313 (2008; Zbl 1171.46023); translation from Mat. Zametki 84, No. 3, 323--333 (2008)], the main result (as it is called by its author) of the paper is stated in Theorem 3. The study of the proof of this theorem revealed that the given ``proof'' contains inherent errors: the aim of the author was to derive the analog of the Minkowski integral inequality for the case of a variable exponent, a function
\(p(x)\), but, actually, the author treats \(p(x)\) as a constant.
In fact, this Theorem 3 is an obvious consequence of Theorem 2.3 given in [\textit{A. R. Schep}, in: Operator theory in function spaces and Banach lattices. Essays dedicated to A. C. Zaanen on the occasion of his 80th birthday. Symposium, Univ. of Leiden, NL, September 1993. Basel: Birkhäuser. 299--308 (1994; Zbl 0849.46020)].Full proof of Kwapień's theorem on representing bounded mean zero functions on \([0,1]\)https://zbmath.org/1472.460272021-11-25T18:46:10.358925Z"Ber, Aleksei"https://zbmath.org/authors/?q=ai:ber.aleksey"Borst, Matthijs"https://zbmath.org/authors/?q=ai:borst.matthijs"Sukochev, Fedor"https://zbmath.org/authors/?q=ai:sukochev.fedor-aThe authors are able to fill a gap in the proof of a theorem in [\textit{S.~Kwapień}, Math. Nachr. 119, 175--179 (1984; Zbl 0575.46003)]. Whereas Kwapień's original proof holds for continuous functions, a gap appears for functions with discontinuities. Indeed, a proof of the following theorem in full generality is given. For every mean zero function \(f \in L_\infty[0, 1]\), there exists \(g \in L_\infty[0,1]\) and a mod \(0\) measure preserving transformation \(T\) of \([0, 1]\) such that \(f=g\circ T - g\). The original gap is discussed and a counterexample is also given.A Pólya-Szegő principle for general fractional Orlicz-Sobolev spaceshttps://zbmath.org/1472.460282021-11-25T18:46:10.358925Z"De Nápoli, Pablo"https://zbmath.org/authors/?q=ai:de-napoli.pablo-luis"Fernández Bonder, Julián"https://zbmath.org/authors/?q=ai:fernandez-bonder.julian"Salort, Ariel"https://zbmath.org/authors/?q=ai:salort.ariel-martinSummary: In this article, we prove modular and norm Pólya-Szegő inequalities in general fractional Orlicz-Sobolev spaces by using the polarization technique. We introduce a general framework which includes the different definitions of these spaces in the literature, and we establish some of its basic properties such as the density of smooth functions. As a corollary, we prove a Rayleigh-Faber-Krahn type inequality for Dirichlet eigenvalues under nonlocal nonstandard growth operators.Completeness and separability of the spaces of variable integrability and summabilityhttps://zbmath.org/1472.460292021-11-25T18:46:10.358925Z"Ghorbanalizadeh, Arash"https://zbmath.org/authors/?q=ai:ghorbanalizadeh.arash-m"Górka, Przemysław"https://zbmath.org/authors/?q=ai:gorka.przemyslawThe variable exponent Lebesgue spaces \(L^{p(\cdot)}\) and the variable exponent mixed Lebesgue-sequence spaces \(l^{q(\cdot)}(L^{p(\cdot)})\) are natural generalizations of the classical Lebesgue spaces \(L^{p}\) and the mixed Lebesgue-sequence spaces \(l^{q}(L^{p})\). They have been studied in, e.g., [\textit{L. Diening} et al., Lebesgue and Sobolev spaces with variable exponents. Berlin: Springer (2011; Zbl 1222.46002); \textit{H. Kempka} and \textit{J. Vybíral}, Proc. Am. Math. Soc. 141, No. 9, 3207--3212 (2013; Zbl 1283.46022)]. From such spaces, one can also investigate derived spaces such as variable exponent Besov spaces, see, e.g., [\textit{A. Almeida} and \textit{P. Hästö}, J. Funct. Anal. 258, No. 5, 1628--1655 (2010; Zbl 1194.46045)].
It is known that for any bounded below measurable function \(p:\mathbb{R}^{n} \to (0, \infty]\) the variable exponent Lebesgue space \(L^{p(\cdot)}\) is a quasi-Banach space. In the present paper, the authors prove that for any pair of bounded below measurable functions \(p,q:\mathbb{R}^{n} \to (0, \infty]\) the variable exponent mixed Lebesgue-sequence spaces \(l^{q(\cdot)}(L^{p(\cdot)})\) are also quasi-Banach spaces. The authors also show, under the assumption of finite essential suprema of \(p\) and \(q\), that the variable exponent mixed Lebesgue-sequence spaces \(l^{q(\cdot)}(L^{p(\cdot)})\) are separable. The separability question was an open problem posed by Hästö in 2017.\( Q_\alpha^{- 1}\) spaces for Hermite operator and their applications in a class of Hermite-dissipative equationshttps://zbmath.org/1472.460302021-11-25T18:46:10.358925Z"Jiao, Ziyun"https://zbmath.org/authors/?q=ai:jiao.ziyun"Huang, Jizheng"https://zbmath.org/authors/?q=ai:huang.jizheng"Li, Pengtao"https://zbmath.org/authors/?q=ai:li.pengtao"Liu, Yu"https://zbmath.org/authors/?q=ai:liu.yuSummary: Let \(H = - \Delta + | x |^2\) be a Hermite operator acting on \(L^2( \mathbb{R}^d)\). In this paper, we introduce a class of the \(Q_\alpha^{- 1}\) spaces associated with \(H\). As an application, with initial data in \(Q_\alpha^{- 1}\) space, we obtain the well-posedness of a class of dissipative equations related to \(H\). In order to get our results, we will prove that the Besov spaces are the real interpolations of the Sobolev spaces associated with \(H\).Integrability properties of integral transforms via Morrey spaceshttps://zbmath.org/1472.460312021-11-25T18:46:10.358925Z"Samko, Natasha"https://zbmath.org/authors/?q=ai:samko.natashaSummary: We show that integrability properties of integral transforms with kernel depending on the product of arguments (which include in particular, popular Laplace, Hankel, Mittag-Leffler transforms and various others) are better described in terms of Morrey spaces than in terms of Lebesgue spaces. Mapping properties of integral transforms of such a type in Lebesgue spaces, including weight setting, are known. We discover that local weighted Morrey and complementary Morrey spaces are very appropriate spaces for describing integrability properties of such transforms. More precisely, we show that under certain natural assumptions on the kernel, transforms under consideration act from local weighted Morrey space to a weighted complementary Morrey space and vice versa, where an interplay between behavior of functions and their transforms at the origin and infinity is transparent. In case of multidimensional integral transforms, for this goal we introduce and use anisotropic mixed norm Morrey and complementary Morrey spaces.Boundedness of localization operators on Lorentz mixed-normed modulation spaceshttps://zbmath.org/1472.460322021-11-25T18:46:10.358925Z"Sandıkçı, Ayşe"https://zbmath.org/authors/?q=ai:sandikci.ayseSummary: In this work we study certain boundedness properties for localization operators on Lorentz mixed-normed modulation spaces, when the operator symbols belong to appropriate modulation spaces, Wiener amalgam spaces, and Lorentz spaces with mixed norms.Martingale inequalities and fractional integral operator in variable Hardy-Lorentz spaceshttps://zbmath.org/1472.460332021-11-25T18:46:10.358925Z"Zeng, Dan"https://zbmath.org/authors/?q=ai:zeng.danSummary: Let \((\Omega, \mathcal{F}, \mathbb{P})\) be a complete probability space. We introduce variable Lorentz space \(\mathcal{L}_{p ( \cdot ) , q}(\Omega)\) defined by rearrangement functions and its related properties. Then, we establish martingale inequalities among these martingale Hardy-Lorentz spaces \(\mathcal{H}_{p ( \cdot ) , q}(\Omega)\) by applying the interpolation theorem. Furthermore, we study the boundedness of the fractional integral operator in variable martingale Hardy spaces \(\mathcal{H}_{p ( \cdot )}^M(\Omega)\) and \(\mathcal{H}_{p ( \cdot ) , q}^M(\Omega)\).Fractional Orlicz-Sobolev embeddingshttps://zbmath.org/1472.460342021-11-25T18:46:10.358925Z"Alberico, Angela"https://zbmath.org/authors/?q=ai:alberico.angela"Cianchi, Andrea"https://zbmath.org/authors/?q=ai:cianchi.andrea"Pick, Luboš"https://zbmath.org/authors/?q=ai:pick.lubos"Slavíková, Lenka"https://zbmath.org/authors/?q=ai:slavikova.lenkaSummary: The optimal Orlicz target space is exhibited for embeddings of fractional-order Orlicz-Sobolev spaces in \(\mathbb{R}^n\). An improved embedding with an Orlicz-Lorentz target space, which is optimal in the broader class of all rearrangement-invariant spaces, is also established. Both spaces of order \(s\in(0, 1)\), and higher-order spaces are considered. Related Hardy type inequalities are proposed as well.A note on Riemann-Liouville fractional Sobolev spaceshttps://zbmath.org/1472.460352021-11-25T18:46:10.358925Z"Carbotti, Alessandro"https://zbmath.org/authors/?q=ai:carbotti.alessandro"Comi, Giovanni E."https://zbmath.org/authors/?q=ai:comi.giovanni-eSummary: Taking inspiration from a recent paper by \textit{M. Bergounioux} et al. [Fract. Calc. Appl. Anal. 20, No. 4, 936--962 (2017; Zbl 1371.26013)], we study the Riemann-Liouville fractional Sobolev space \(W^{s,p}_{RL,a+}(I)\), for \(I=(a,b)\) for some \(a,b\in\mathbb{R}\), \(a<b\), \(s\in(0,1)\) and \(p\in [1,\infty]\); that is, the space of functions \(u \in L^p(I)\) such that the left Riemann-Liouville \((1-s)\)-fractional integral \(I_{a+}^{1-s}[u]\) belongs to \(W^{1,p}(I)\). We prove that the space of functions of bounded variation \(BV(I)\) and the fractional Sobolev space \(W^{s,1}(I)\) continuously embed into \(W^{s,1}_{RL,a+}(I)\). In addition, we define the space of functions with left Riemann-Liouville \(s\)-fractional bounded variation, \(BV^s_{RL,a+}(I)\), as the set of functions \(u\in L^1(I)\) such that \(I^{1-s}_{a+}[u]\in BV(I)\), and we analyze some fine properties of these functions. Finally, we prove some fractional Sobolev-type embedding results and we analyze the case of higher order Riemann-Liouville fractional derivatives.Erratum to: ``Orlicz norm and Sobolev-Orlicz capacity on ends of tree based on probabilistic Bessel kernels''https://zbmath.org/1472.460362021-11-25T18:46:10.358925Z"Hara, C."https://zbmath.org/authors/?q=ai:hara.chiaki|hara.carmem"Iijima, R."https://zbmath.org/authors/?q=ai:iijima.ryota"Kaneko, H."https://zbmath.org/authors/?q=ai:kaneko.hiroyuki|kaneko.hiromichi|kaneko.hideaki|kaneko.hajime|kaneko.hiromi|kaneko.hiroshi|kaneko.hiroki|kaneko.haruhiko"Matsumoto, H."https://zbmath.org/authors/?q=ai:matsumoto.hideyuki|matsumoto.hironori|matsumoto.hiroaki|matsumoto.hirotaka|matsumoto.hisayoshi|matsumoto.hideya|matsumoto.hideki|matsumoto.hiroshi|matsumoto.hisaaki|matsumoto.hiroyuki.1From the text: The notation \(\varphi(t)\) in the integrands of the statements of Proposition 6.2, Theorem 6.3, Theorem 6.4 and in the proof of Theorem 6.4 in the authors' paper [ibid. 7, No. 1, 24--38 (2015; Zbl 1341.46023)] must be replaced by \(\varphi(1/t)\).On the regularity of characteristic functionshttps://zbmath.org/1472.460372021-11-25T18:46:10.358925Z"Sickel, Winfried"https://zbmath.org/authors/?q=ai:sickel.winfriedSummary: In this survey we shall deal with the regularity of characteristic functions \(\mathcal{X}_E\) of subsets \(E\) of \(\mathbb{R}^d\) in the framework of Besov spaces. We will describe a number of necessary and sufficient conditions to guarantee membership in a Besov space of given smoothness \(s\) and with integrability \(p\). Several examples are discussed in detail.
For the entire collection see [Zbl 1457.35004].On compact subsets of Sobolev spaces on manifoldshttps://zbmath.org/1472.460382021-11-25T18:46:10.358925Z"Skrzypczak, Leszek"https://zbmath.org/authors/?q=ai:skrzypczak.leszek"Tintarev, Cyril"https://zbmath.org/authors/?q=ai:tintarev.kyrilSummary: The paper considers compactness of Sobolev embeddings of non-compact manifolds, restricted to subsets (typically subspaces) defined either by conditions of symmetry (or quasisymmetry) relative to actions of compact groups, or by restriction in the number of variables, i.e. consisting of functions of the form \(f\circ \varphi\) with a fixed \(\varphi \). The manifolds are assumed to satisfy general common conditions under which Sobolev embeddings exist. We provide sufficient conditions for compactness of the embeddings, which in many situations are also necessary.A truncated real interpolation method and characterizations of screened Sobolev spaceshttps://zbmath.org/1472.460392021-11-25T18:46:10.358925Z"Stevenson, Noah"https://zbmath.org/authors/?q=ai:stevenson.noah"Tice, Ian"https://zbmath.org/authors/?q=ai:tice.ianSummary: In this paper we prove structural and topological characterizations of the screened Sobolev spaces with screening functions bounded below and above by positive constants. We generalize a method of interpolation to the case of seminormed spaces. This method, which we call the truncated method, generates the screened Sobolev subfamily and a more general screened Besov scale. We then prove that the screened Besov spaces are equivalent to the sum of a Lebesgue space and a homogeneous Sobolev space and provide a Littlewood-Paley frequency space characterization.On Trudinger-type inequalities in Orlicz-Morrey spaces of an integral formhttps://zbmath.org/1472.460402021-11-25T18:46:10.358925Z"Hurri-Syrjänen, Ritva"https://zbmath.org/authors/?q=ai:hurri-syrjanen.ritva"Ohno, Takao"https://zbmath.org/authors/?q=ai:ohno.takao"Shimomura, Tetsu"https://zbmath.org/authors/?q=ai:shimomura.tetsuSummary: We give Trudinger-type inequalities for Riesz potentials of functions in Orlicz-Morrey spaces of an integral form over non-doubling metric measure spaces. Our results are new even for the doubling metric measure setting. In particular, our results improve and extend the previous results in Morrey spaces of an integral form in the Euclidean case.Classic and exotic Besov spaces induced by good gridshttps://zbmath.org/1472.460412021-11-25T18:46:10.358925Z"Smania, Daniel"https://zbmath.org/authors/?q=ai:smania.danielSummary: In a previous work we introduced Besov spaces \(\mathcal{B}^s_{p,q}\) defined on a measure space with a good grid, with \(p\in[1,\infty)\), \(q\in[1,\infty]\) and \(0< s<1/p\). Here we show that classical Besov spaces on compact homogeneous spaces are examples of such Besov spaces. On the other hand we show that even Besov spaces defined by a good grid made of partitions by intervals may differ from a classical Besov space, giving birth to exotic Besov spaces.Maximal factorization of operators acting in Köthe-Bochner spaceshttps://zbmath.org/1472.460422021-11-25T18:46:10.358925Z"Calabuig, J. M."https://zbmath.org/authors/?q=ai:calabuig.jose-m"Fernández-Unzueta, M."https://zbmath.org/authors/?q=ai:fernandez-unzueta.maite"Galaz-Fontes, F."https://zbmath.org/authors/?q=ai:galaz-fontes.fernando"Sánchez-Pérez, E. A."https://zbmath.org/authors/?q=ai:sanchez-perez.enrique-alfonsoSummary: Using some representation results for Köthe-Bochner spaces of vector valued functions by means of vector measures, we analyze the maximal extension for some classes of linear operators acting in these spaces. A factorization result is provided, and a specific representation of the biggest vector valued function space to which the operator can be extended is given. Thus, we present a generalization of the optimal domain theorem for some types of operators on Banach function spaces involving domination inequalities and compactness. In particular, we show that an operator acting in Bochner spaces of \(p\)-integrable functions for any \(1<p<\infty\) having a specific compactness property can always be factored through the corresponding Bochner space of 1-integrable functions. Some applications in the context of the Fourier type of Banach spaces are also given.Nuclear global spaces of ultradifferentiable functions in the matrix weighted settinghttps://zbmath.org/1472.460432021-11-25T18:46:10.358925Z"Boiti, Chiara"https://zbmath.org/authors/?q=ai:boiti.chiara"Jornet, David"https://zbmath.org/authors/?q=ai:jornet.david"Oliaro, Alessandro"https://zbmath.org/authors/?q=ai:oliaro.alessandro"Schindl, Gerhard"https://zbmath.org/authors/?q=ai:schindl.gerhardThe authors show that the Hermite functions are a Schauder basis of many global weighted spaces of ultradifferentiable functions, thus extending the previous work by \textit{M. Langenbruch} [Manuscr. Math. 119, No. 3, 269--285 (2006; Zbl 1101.46026)]. Moreover, they determine the coefficient spaces corresponding to this Hermite expansion. These results are applied to spaces defined by weight functions \(\mathcal{S}_{[\omega]}(\mathbb{R}^d)\), with \([\omega] = (\omega)\) (Beurling setting) or \([\omega] = \{\omega\}\) (Roumieu setting). Therefore, they extend part of the previous work of \textit{J.-M. Aubry} [J. Lond. Math. Soc., II. Ser. 78, No. 2, 392--406 (2008; Zbl 1170.46007)] to the several variables case. As a consequence, the authors are able to generalize their previous study about the nuclearity of the space \(\mathcal{S}_{(\omega)}(\mathbb{R}^d)\) to global spaces of ultradifferentiable functions defined by weight matrices. In particular, the authors characterize in a very general way when the Hermite functions are contained in the classes considered in the paper and this fact is closely related to classes being non-trivial. Indeed, they deduce from their results that, in the Beurling setting, the space \(\mathcal{S}_{(\omega)}(\mathbb{R}^d)\) contains the Hermite functions if and only if \(\omega(t) = o(t^2)\) as \(t\) tends to infinity. In the same way, in the Roumieu case, the space \(\mathcal{S}_{\{\omega\}}(\mathbb{R}^d)\) contains the Hermite functions if and only if \(\omega(t) = O(t^2)\) as \(t\) tends to infinity.Nuclearity of rapidly decreasing ultradifferentiable functions and time-frequency analysishttps://zbmath.org/1472.460442021-11-25T18:46:10.358925Z"Boiti, Chiara"https://zbmath.org/authors/?q=ai:boiti.chiara"Jornet, David"https://zbmath.org/authors/?q=ai:jornet.david"Oliaro, Alessandro"https://zbmath.org/authors/?q=ai:oliaro.alessandro"Schindl, Gerhard"https://zbmath.org/authors/?q=ai:schindl.gerhardThis article completes the study begun by the first three authors in their paper [in: Advances in microlocal and time-frequency analysis. Contributions of the conference on microlocal and time-frequency analysis 2018, MLTFA18, in honor of Prof. Luigi Rodino on the occasion of his 70th birthday, Torino, Italy, July 2--6, 2018. Cham: Birkhäuser. 121--129 (2020; Zbl 1457.46052)]. The authors use techniques from time-frequency analysis to show that the space \(S_{\omega}\) of rapidly decreasing ultradifferentiable functions is nuclear for every weight function \(\omega\) such that \(\omega(t) = o(t)\) as \(t\) goes to infinity. Moreover, they show that for a sequence \((M_p)\) satisfying the classical condition \((M1)\) of Komatsu, the Beurling space \(S_{(M_p)}\) when defined with \(L_2\)-norms is nuclear if and only if \((M_p)\) satisfies condition \((M2)'\) of Komatsu. The present research has been continued and extended by the authors in [Banach J. Math. Anal. 15, No. 1, Paper No. 14, 38 p. (2021; Zbl 1472.46043)]. Related work has been published by \textit{A. Debrouwere} et al. [Proc. Am. Math. Soc. 148, No. 12, 5171--5180 (2020; Zbl 07268384); Collect. Math. 72, No. 1, 203--227 (2021; Zbl 1465.46003)].Asymptotically almost periodic and asymptotically almost automorphic vector-valued generalized functionshttps://zbmath.org/1472.460452021-11-25T18:46:10.358925Z"Kostić, Marko"https://zbmath.org/authors/?q=ai:kostic.markoSummary: The main purpose of this paper is to introduce the notion of an asymptotically almost periodic ultradistribution and asymptotically almost automorphic ultradistribution with values in a Banach space, as well as to further analyze the classes of asymptotically almost periodic and asymptotically almost automorphic distributions with values in a Banach space. We provide some applications of the introduced concepts in the analysis of systems of ordinary differential equations.Characterizations of continuous operators on \(C_b(X)\) with the strict topologyhttps://zbmath.org/1472.460462021-11-25T18:46:10.358925Z"Nowak, Marian"https://zbmath.org/authors/?q=ai:nowak.marian"Stochmal, Juliusz"https://zbmath.org/authors/?q=ai:stochmal.juliuszSummary: Let \(X\) be a completely regular Hausdorff space and \(C_b(X)\) be the space of all bounded continuous functions on \(X\), equipped with the strict topology \(\beta \). We study some important classes of \((\beta,\Vert\cdot\Vert_E)\)-continuous linear operators from \(C_b(X)\) to a Banach space \((E,\Vert\cdot\Vert_E)\): \(\beta\)-absolutely summing operators, compact operators and \(\beta\)-nuclear operators. We characterize compact operators and \(\beta\)-nuclear operators in terms of their representing measures. It is shown that dominated operators and \(\beta\)-absolutely summing operators \(T:C_b(X)\rightarrow E\) coincide and if, in particular, \(E\) has the Radon-Nikodym property, then \(\beta\)-absolutely summing operators and \(\beta\)-nuclear operators coincide. We generalize the classical theorems of Pietsch, Tong and Uhl concerning the relationships between absolutely summing, dominated, nuclear and compact operators on the Banach space \(C(X)\), where \(X\) is a compact Hausdorff space.Schauder bases, LUR Banach spaces and diametrically with empty interiorhttps://zbmath.org/1472.460472021-11-25T18:46:10.358925Z"Budzynska, Monika"https://zbmath.org/authors/?q=ai:budzynska.monika"Kaczor, Wieslawa"https://zbmath.org/authors/?q=ai:kaczor.wieslawa-j"Kot, Mariola"https://zbmath.org/authors/?q=ai:kot.mariola"Kuczumow, Tadeusz"https://zbmath.org/authors/?q=ai:kuczumow.tadeuszSummary: In this paper we prove that if a reflexive Banach space \((X,\|\cdot\|)\) has a Schauder basis, then it has an equivalent norm \(\|\cdot\|_0\) such that the Banach space \((X,\|\cdot\|)_0)\) is LUR and has a diametrically complete set with empty interior. This result extends the Maluta theorem about existence of such a set in \(l^2\) with the Day norm.Multipliers of modules of continuous vector-valued functionshttps://zbmath.org/1472.460482021-11-25T18:46:10.358925Z"Khan, Liaqat Ali"https://zbmath.org/authors/?q=ai:khan.liaqat-ali"Alsulami, Saud M."https://zbmath.org/authors/?q=ai:alsulami.saud-mastour-aSummary: In [Pac. J. Math. 11, 1131--1149 (1961; Zbl 0127.33302)], \textit{J.-K. Wang} showed that if \(A\) is the commutative \(C^*\)-algebra \(C_0(X)\) with \(X\) a locally compact Hausdorff space, then \(M(C_0(X)) \cong C_b(X)\). Later, this type of characterization of multipliers of spaces of continuous scalar-valued functions has also been generalized to algebras and modules of continuous vector-valued functions by several authors. In this paper, we obtain further extension of these results by showing that \[\text{H} \text{o} \text{m}_{C_0(X, A)}(C_0(X, E), C_0(X, F)) \simeq C_{s, b}(X, \text{H} \text{o} \text{m}_A(E, F)),\] where \(E\) and \(F\) are \(p\)-normed spaces which are also essential isometric left \(A\)-modules with \(A\) being a certain commutative \(F\)-algebra, not necessarily locally convex. Our results unify and extend several known results in the literature.A weaker Gleason-Kahane-Żelazko theorem for modules and applications to Hardy spaceshttps://zbmath.org/1472.460492021-11-25T18:46:10.358925Z"Sebastian, Geethika"https://zbmath.org/authors/?q=ai:sebastian.geethika"Daniel, Sukumar"https://zbmath.org/authors/?q=ai:daniel.sukumarLet \(A\) be a complex unital Banach algebra. The paper deals with the question of finding sufficient conditions under which a map \(\Lambda:M\rightarrow\mathbb C\) between a left \(A\)-module \(M\) and the field \(\mathbb C\) is linear.\par Let \(A\) be a complex unital Banach algebra, \(M\) a left \(A\)-module, \(\Lambda:M\rightarrow\mathbb C\) a non-zero map and \(S\) a non-empty subset of \(M\), satisfying the following conditions:
\begin{itemize}
\item[(S1)]
\(\theta_M\not\in S\) and \(S\) generates \(M\) as an \(A\)-module;
\item[(S2)]
if \(a\) is invertible in \(A\) and \(s\in S\), then \(as\in S\);
\item[(S3)]
for all \(s_1,s_2\in S\), there exist \(a_1,a_2\in A\) such that \(a_1s_1=a_2s_2\in S\).
\end{itemize}
The main results of the paper are the following:
\begin{itemize}
\item[(1)]
If \(\Lambda\) is an \(\mathbb R\)-linear functional on \(M\) such that \(\Lambda(m)s-\Lambda(s)m\not\in S\) for all \(m\in M\) and \(s\in S\), then \(\Lambda\) is \(\mathbb C\)-linear and there exists a unique character \(\chi:A\rightarrow\mathbb C\) such that \(\Lambda(am)=\chi(a)\Lambda(m)\) for all \(a\in A\) and \(m\in M\).
\item[(2)]
If \(\Lambda(\theta_M)=0\) and \((\Lambda(m_1)-\Lambda(m_2))s-(m_1-m_2)\Lambda(s)\not\in S\) for all \(m_1,m_2\in M\) and \(s\in S\), then there exists a unique character \(\chi:A\rightarrow\mathbb C\) such that \(\Lambda(as)=\chi(a)\Lambda(s)\) for all \(a\in A\) and \(s\in S\).
\item[(3)]
If, in addition, \[\sum_{j=1}^n\Lambda(s_j)=\Lambda\Biggl(\sum_{j=1}^ns_j\Biggr)\] for all \(n\in {\mathbb Z}^+\) and \(s_1,\dots,s_n\in S\), then \(\Lambda\) is linear.
\item[(4)]
If \(\chi:A\rightarrow\mathbb C\) is a map such that \(\Lambda(am)=\chi(a)\Lambda(m)\) and \(\Lambda(m)s-\Lambda(s)m\not\in S\) for all \(a\in A$, $m\in M\) and \(s\in S\), then \(\chi\) is linear if and only if the map \(\tau_a:{\mathbb C}\rightarrow\mathbb C\), defined as \(\tau_a(\lambda)=\chi(\lambda e_A-a)\) for each \(\lambda\in\mathbb C\), is an entire function on \(\mathbb C\) for every \(a\in A\).
\end{itemize}
In the final chapter, the results of this type are also applied to study the properties of some Hardy spaces.Extensions of uniform algebrashttps://zbmath.org/1472.460502021-11-25T18:46:10.358925Z"Morley, Sam"https://zbmath.org/authors/?q=ai:morley.samCole's counterexample to the peak point conjecture
[\textit{B.~J. Cole}, One-point parts and the peak point conjecture. Ph.D. dissertation, Yale University, New Haven, CT (1968)]
provides a construction of extensions of uniform algebras having various desirable properties, and has been used in the theory of uniform algebras (e.g., [\textit{J.~F. Feinstein}, Stud. Math. 148, No.~1, 67--74 (2001; Zbl 1055.46035); Proc. Am. Math. Soc. 132, No.~8, 2389--2397 (2004; Zbl 1055.46036)]).
The paper under review introduces some new classes of uniform algebra extensions and investigates the properties that are inherited by any uniform algebra extensions such as being nontrivial, natural, regular and normal. The relationship between peak sets in the weak sense and the Shilov boundary of an extension to those of the original uniform algebra is studied as well.
Let \(X\) and \(Y\) be compact Hausdorff spaces, let \(A\) be a uniform algebra on \(X\) and \(B\) be a uniform algebra on \(Y\). If there exists a continuous surjection \(\Pi: Y \to X\) such that \(\Pi^*(A) \subseteq B\), then \(B\) is called a uniform algebra extension of \(A\). The author introduces a class of uniform algebra extensions called generalised Cole extensions, namely, if there exists a continuous surjection \(\Pi: Y \to X\) and a unital linear map \(T: C(X) \to C(Y)\) with \(\|T\|= 1\) such that \(T\circ \Pi^*= \mathrm{ id}_{C(X)}\), \(\Pi^*(A) \subseteq B\) and \(T(B)= A\). In this case, \(\Pi\) and \(T\) are called associated maps to the extension. It is shown in the paper, Section 5, that generalised Cole extensions preserve several properties of the original uniform algebra. The author also studies generalised Cole extensions implemented by a compact group. In general, a generalised Cole extension need not be implemented by a group; however, if the associated maps to the extension \(B\) satisfy \(\|\mathrm{id}_{C(Y)}- \Pi^*\circ T\|= 1\), then there exists a finite group \(G\) such that \(B\) is a generalised Cole extension implemented by \(G\). The paper is concluded with some examples and two open questions.Analytic structure on the spectrum of the algebra of symmetric analytic functions on \(L_\infty \)https://zbmath.org/1472.460512021-11-25T18:46:10.358925Z"Galindo, Pablo"https://zbmath.org/authors/?q=ai:galindo.pablo"Vasylyshyn, Taras"https://zbmath.org/authors/?q=ai:vasylyshyn.taras-v"Zagorodnyuk, Andriy"https://zbmath.org/authors/?q=ai:zagorodnyuk.andriy-vLet \(H_{bs}(L_\infty)\) denote the Fréchet algebra of all entire symmetric functions \(f:L_\infty\to\mathbb{C}\) which are bounded on bounded subsets of \(L_\infty\), endowed with the topology of uniform convergence on bounded sets. The authors prove that \(H_{bs}(L_\infty)\) is isomorphic to \(H(H(\mathbb{C})^*_\beta)\), the algebra of all analytic functions on the (DFM)-space \(H(\mathbb{C})^*_\beta\). Thus the spectrum of \(H_{bs}(L_\infty)\), after its identification with \(H(\mathbb{C})^*\), can be endowed with the analytic structure provided by \(H(\mathbb{C})^*_\beta\).
In his thesis, E.~A. Michael posed the problem of whether every complex-valued homomorphism on a complex commutative Fréchet algebra is necessarily continuous. \textit{D. Clayton} [Rocky Mt. J. Math. 5, 337--344 (1975; Zbl 0325.46055)], \textit{M. Schottenloher} [Arch. Math. 37, 241--247 (1981; Zbl 0471.46036)] and \textit{J. Mujica} [Complex analysis in Banach spaces. Holomorphic functions and domains of holomorphy in finite and infinite dimensions. Amsterdam/New York/Oxford: North-Holland (1986; Zbl 0586.46040)] constructed concrete Fréchet algebras $A$ with the property that the Michael problem can be solved in general if and only if it can be solved for such an algebra~$A$. From the above result, the authors of the paper under review show in an elementary way that also \(H_{bs}(L_\infty)\) is a test algebra for the Michael problem.On the covariance of Moore-Penrose inverses in rings with involutionhttps://zbmath.org/1472.460522021-11-25T18:46:10.358925Z"Mahzoon, Hesam"https://zbmath.org/authors/?q=ai:mahzoon.hesamSummary: We consider the so-called covariance set of Moore-Penrose inverses in rings with an involution. We deduce some new results concerning covariance set. We will show that if \(a\) is a regular element in a \(C^\ast\)-algebra, then the covariance set of \(a\) is closed in the set of invertible elements (with relative topology) of \(C^\ast\)-algebra and is a cone in the \(C^\ast\)-algebra.Involutive operator algebrashttps://zbmath.org/1472.460532021-11-25T18:46:10.358925Z"Blecher, David P."https://zbmath.org/authors/?q=ai:blecher.david-p"Wang, Zhenhua"https://zbmath.org/authors/?q=ai:wang.zhenhuaLet \(\mathcal{H}\) be a complex Hilbert space and \(\mathcal{A}\) be an operator algebra on \(\mathcal{H}\), that is, a closed subalgebra of \(\mathcal{B}(\mathcal{H})\). When \(\mathcal{A}\) is equipped with operator space norms inherited from \(\mathcal{B}(\mathcal{H})\) and a completely isometric algebra involution \(\dagger\), it is called an operator \(*\)-algebra. There are four types of involutions (bijections of period two) on an operator algebra, including the above one as the first type. In the paper under review, the authors investigate the structure of operator \(*\)-algebras and remark that their results are applicable to algebras with other types of involution, after appropriate modifications in the arguments.
The paper begins with establishing some characterisations of operator algebras and operator \(*\)-algebras \(\mathcal{A}\), in terms of their biggest \(C^*\)-cover \(C^*_{\max}(\mathcal{A})\) and smallest \(C^*\)-cover \(C^*_e(\mathcal{A})\) of \(\mathcal{A}\).
In the third section of the paper, a wide range of examples of operator \(*\)-algebras are presented, including some uniform algebras, operator algebras generated by a single element, and operator algebras obtained from operator systems by Paulsen's off diagonal technique. Then several characterisations of operator \(*\)-algebras with contractive approximate identities and, in particular, those with countable contractive approximate identities are given. Moreover, analogues of the Arens-Kadison theorem and Cohen's factorisation theorem for operator \(*\)-algebras are proved.
The last section of the paper is devoted to hereditary subalgebras of operator \(*\)-algebras and their relations with Akemann's noncommutative topology. To be more precise, suppose that \(\mathcal{A}\) with the involution \(\dagger\) turns into an operator \(*\)-algebra. An orthogonal projection \(p\in\mathcal{A}^{**}\) is open in \(\mathcal{A}^{**}\) if there is a net \((x_t)\) in \(\mathcal{A}\) such that \[ x_t=px_t=x_tp\overset{w^*}\longrightarrow p. \] If, moreover, \(p=p^\dagger\), then we say that \(p\) is \(\dagger\)-open. In this case, the closed \(\dagger\)-subalgebra \(\mathcal{D}=p\mathcal{A}^{**}p\cap \mathcal{A}\) is called a \(\dagger\)-hereditary subalgebra of \(\mathcal{A}\) and \(p\) is called the support projection of \(\mathcal{D}\). The authors characterise \(\dagger\)-hereditary subalgebras in terms of one-sided ideals, analogous to a well-known characterisation of hereditary \(C^*\)-subalgebras. Besides, for every \(\dagger\)-hereditary subalgebra \(\mathcal{B}\) of \(\mathcal{A}\) a set \(E\) of real positive elements of \(\mathcal{A}\) is identified which generates \(\mathcal{B}\), that is, \(\mathcal{B}=\overline{E\mathcal{A} E}\). In particular separable \(\dagger\)-hereditary subalgebras or \(\dagger\)-hereditary subalgebras with countable contractive approximate identities are of the form \(\overline{x\mathcal{A} x}\) for certain \(x\in\mathcal{A}\).
A projection \(q\in\mathcal{A}^{**}\) is called \(\dagger\)-compact in \(\mathcal{A}^{**}\) if \(1-q\) is open in \(\mathcal{A}^{**}\) and there is an \(x\) in the unit ball of \(\mathcal{A}\) such that \(q=qx\). The paper is concluded with characterisations of compact projections and noncommutative analogues of Urysohn's lemma and Tietze's extension theorem.A multiplicative Gleason-Kahane-Żelazko theorem for \(C^\star \)-algebrashttps://zbmath.org/1472.460542021-11-25T18:46:10.358925Z"Brits, R."https://zbmath.org/authors/?q=ai:brits.rudi-m"Mabrouk, M."https://zbmath.org/authors/?q=ai:mabrouk.mohamed|mabrouk.mongi|mabrouk.mai-s|mabrouk.mohammed"Touré, C."https://zbmath.org/authors/?q=ai:toure.cheickSummary: If \(\mathfrak{A}\) is a unital complex Banach algebra, and if \(\sigma(a)\) denotes the spectrum of an element \(a \in \mathfrak{A} \), then the famous Gleason-Kahane-Żelazko theorem says that any linear functional \(\phi : \mathfrak{A} \to \mathbb{C}\) satisfying \(\phi(a) \in \sigma (a)\) for each \(a \in \mathfrak{A} \), is multiplicative and continuous. In this paper we establish a multiplicative Gleason-Kahane-Żelazko theorem for the case where \(\mathfrak{A}\) is a \(C^\star \)-algebra. Specifically, if \(\mathfrak{A}\) is a \(C^\star \)-algebra, then any continuous multiplicative functional \(\phi : \mathfrak{A} \to \mathbb{C}\) satisfying \(\phi(a) \in \sigma (a)\) for each \(a \in \mathfrak{A} \), is linear and hence a character of \(\mathfrak{A} \).Remarks on the \(K\)-theory of \(C^*\)-algebras of products of odometershttps://zbmath.org/1472.460552021-11-25T18:46:10.358925Z"Li, Hui"https://zbmath.org/authors/?q=ai:li.hui.3|li.hui.4|li.hui.2|li.hui|li.hui.1|li.hui.5Summary: We pose a conjecture on the \(K\)-theory of the self-similar \(k\)-graph \(C\)*-algebra of a standard product of odometers. We generalize the \(C^*\)-algebra \(\mathcal{Q}_S\) to any subset of \(\mathbb{N}^\times\setminus\{1\}\) and then realize it as the self-similar \(k\)-graph \(C^*\)-algebra of a standard product of odometers.\(\mathrm C^*\)-algebras of right LCM one-relator monoids and Artin-Tits monoids of finite typehttps://zbmath.org/1472.460562021-11-25T18:46:10.358925Z"Li, Xin"https://zbmath.org/authors/?q=ai:li.xin.13|li.xin.15|li.xin.14|li.xin.1|li.xin.5|li.xin.10|li.xin.3|li.xin.12|li.xin.11|li.xin.6|li.xin.4|li.xin.9|li.xin.7|li.xin.2|li.xin"Omland, Tron"https://zbmath.org/authors/?q=ai:omland.tron-a"Spielberg, Jack"https://zbmath.org/authors/?q=ai:spielberg.jackSummary: We study \(\mathrm C^*\)-algebras generated by left regular representations of right LCM one-relator monoids and Artin-Tits monoids of finite type. We obtain structural results concerning nuclearity, ideal structure and pure infiniteness. Moreover, we compute \(K\)-theory. Based on our \( K\)-theory results, we develop a new way of computing \( K\)-theory for certain group \(\mathrm C^*\)-algebras and crossed products.The uniform Roe algebra of an inverse semigrouphttps://zbmath.org/1472.460572021-11-25T18:46:10.358925Z"Lledó, Fernando"https://zbmath.org/authors/?q=ai:lledo.fernando"Martínez, Diego"https://zbmath.org/authors/?q=ai:martinez.diego-c|martinez.diego-rSummary: Given a discrete and countable inverse semigroup \(S\) one can study, in analogy to the group case, its geometric aspects. In particular, we can equip \(S\) with a natural metric, given by the path metric in the disjoint union of its Schützenberger graphs. This graph, which we denote by \(\Lambda_S\), inherits much of the structure of \(S\). In this article we compare the \(C^*\)-algebra \(\mathcal{R}_S\), generated by the left regular representation of \(S\) on \(\ell^2(S)\) and \(\ell^\infty(S)\), with the uniform Roe algebra over the metric space, namely \(C_u^\ast( \Lambda_S)\). This yields a characterization of when \(\mathcal{R}_S = C_u^\ast( \Lambda_S)\), which generalizes finite generation of \(S\). We have termed this by \textit{admitting a finite labeling} (or being FL), since it holds when \(\Lambda_S\) can be labeled in a finitary manner.
The graph \(\Lambda_S\), and the FL condition, also allow to analyze large scale properties of \(\Lambda_S\) and relate them with \(C^*\)-properties of the uniform Roe algebra. In particular, we show that domain measurability of \(S\) (a notion generalizing Day's definition of amenability of a semigroup, cf. [\textit{P.~Ara} et al., J. Funct. Anal. 279, No.~2, Article ID 108530, 43 p. (2020; Zbl 1446.46035)]) is a quasi-isometric invariant of \(\Lambda_S\). Moreover, we characterize property A of \(\Lambda_S\) (or of its components) in terms of the nuclearity and exactness of the corresponding \(C^*\)-algebras. We also treat the special classes of \(F\)-inverse and \(E\)-unitary inverse semigroups from this large scale point of view.Erratum to: ``\(C^\ast\)-simplicity of locally compact powers groups''https://zbmath.org/1472.460582021-11-25T18:46:10.358925Z"Raum, Sven"https://zbmath.org/authors/?q=ai:raum.svenErratum to the author's paper [ibid. 748, 173--205 (2019; Zbl 1419.46035)].
An error in Lemma~5.1 of that paper is reported. It is further shown that still
Theorem~G persists, while Theorem~B is not correct in the stated generality.Unbounded operator algebrashttps://zbmath.org/1472.460592021-11-25T18:46:10.358925Z"Asadi, Mohammad B."https://zbmath.org/authors/?q=ai:asadi.mohammad-b"Hassanpour-Yakhdani, Z."https://zbmath.org/authors/?q=ai:hassanpour-yakhdani.zahra"Shamloo, S."https://zbmath.org/authors/?q=ai:shamloo.saraSummary: In this paper, we introduce the notion of abstract local operator algebras and operator modules, and provide a representation theorem for them which extends the BRS theorem for operator algebras. Furthermore, we give a new proof for the representation theorem of local operator systems. Also, we investigate the Haagerup tensor product of local operator spaces and Morita equivalence of local operator algebras.Kolmogorov decomposition of conditionally completely positive definite kernelshttps://zbmath.org/1472.460602021-11-25T18:46:10.358925Z"Ghaemi, Mostafa"https://zbmath.org/authors/?q=ai:ghaemi.mostafa"Moslehian, M. S."https://zbmath.org/authors/?q=ai:moslehian.mohammad-sal"Xu, Qingxiang"https://zbmath.org/authors/?q=ai:xu.qingxiangThe authors study conditionally completely positive definite kernels taking values in the algebra of bounded self-maps of a \(C^*\)-algebra. The main result is the Kolmogorov decomposition for conditionally completely positive definite kernels satisfying certain algebraic properties.
The authors also present a characterization of conditionally completely positive definite kernels majorized by a kernel under some mild conditions, and find a sufficient condition for the existence of an extreme point of a convex set consisting of some special kernels.Schatten classes for Hilbert modules over commutative \(C^\ast \)-algebrashttps://zbmath.org/1472.460612021-11-25T18:46:10.358925Z"Stern, Abel B."https://zbmath.org/authors/?q=ai:stern.abel-b"van Suijlekom, Walter D."https://zbmath.org/authors/?q=ai:van-suijlekom.walter-danielThe authors introduce the notion of \textit{Schatten classes} of adjointable operators on Hilbert \(C^*\)-modules over \textit{abelian} \(C^*\)-algebras. These Schatten classes form a two-sided ideal and are equipped with a Banach norm and a \(C^*\)-valued trace with the expected properties. For trivial Hilbert bundles, the Schatten class can be identified with Schatten-norm-continuous maps from the base space into the Schatten class on the Hilbert space fiber. As applications, the \(C^*\)-valued Fredholm determinant, operator zeta function, and \(p\)-summable unbounded Kasparov cycles are introduced in the commutative setting.Maximality and finiteness of type 1 subdiagonal algebrashttps://zbmath.org/1472.460622021-11-25T18:46:10.358925Z"Ji, Guoxing"https://zbmath.org/authors/?q=ai:ji.guoxingSummary: Let \(\mathfrak{A}\) be a type 1 subdiagonal algebra in a \(\sigma\)-finite von Neumann algebra \(\mathcal{M}\) with respect to a faithful normal conditional expectation \(\Phi\). We give necessary and sufficient conditions for which \(\mathfrak{A}\) is maximal among the \(\sigma\)-weakly closed subalgebras of \(\mathcal{M}\). In addition, we show that a type 1 subdiagonal algebra in a finite von Neumann algebra is automatically finite which gives a positive answer of \textit{W. B. Arveson}'s finiteness problem in [Am. J. Math. 89, 578--642 (1967; Zbl 0183.42501)] in type 1 case.Relative commutants of finite groups of unitary operators and commuting mapshttps://zbmath.org/1472.460632021-11-25T18:46:10.358925Z"Magajna, Bojan"https://zbmath.org/authors/?q=ai:magajna.bojanSummary: For a von Neumann algebra \(\mathcal{R}\) we determine the commutant of the set \(\{u \otimes u : u \in \mathcal{R},\, u \text{ unitary} \}\) and normal functionals on \(\mathcal{R} \overline{\otimes} \mathcal{R}\) that are invariant under all automorphisms implemented by \(u \otimes u\) for \(u\) unitary in \(\mathcal{R} \). For a finite group \(G\) of unitary operators on a Hilbert space \(\mathcal{H}\) implementing automorphisms of a von Neumann algebra \(\mathcal{S} \subseteq B(\mathcal{H})\) we describe the relative commutant of \(\mathcal{S}\) in the von Neumann algebra generated by \(\mathcal{S}\) and \(G\).Central sequences in subhomogeneous unital \(\mathrm{C}^\ast\)-algebrashttps://zbmath.org/1472.460642021-11-25T18:46:10.358925Z"Hadwin, Don"https://zbmath.org/authors/?q=ai:hadwin.donald-w"Pendharkar, Hemant"https://zbmath.org/authors/?q=ai:pendharkar.hemantSummary: Suppose that \(\mathcal{A}\) is a unital subhomogeneous \(\mathrm{C}^\ast\)-algebra. We show that every central sequence in \(\mathcal{A}\) is hypercentral if and only if every pointwise limit of a sequence of irreducible representations is multiplicity free. We also show that every central sequence in \(\mathcal{A}\) is trivial if and only if every pointwise limit of irreducible representations is irreducible. Finally, we give a nice representation of the latter algebras.Lyapunov convexity theorem for von Neumann algebras and Jordan operator structureshttps://zbmath.org/1472.460652021-11-25T18:46:10.358925Z"Hamhalter, Jan"https://zbmath.org/authors/?q=ai:hamhalter.janBy noncommutative Lyapunov type theorem for von Neumann algebras, the author means sufficient conditions on a von Neumann algebra \(M\), a linear space \(X\) and a finitely additive measure \(\mu\) on the projection lattice \(P(M)\) with values in \(X\), ensuring that the range \(\mu(P(M))\) is a convex subset of \(X\). The starting point is a reformulation of the classical Lyapunov theorem, where \(M\) is abelian and non-atomic, \(X=\mathbb{C}^n\) and \(\mu\) is bounded. In case \(\mu\) has an affine extension \(\hat\mu\) to the positive part \(M_1^+\) of the unit ball of \(M\), a stronger formulation is the claim that the range is identical to \(\hat\mu(M_1^+)\). The author proves a theorem of this kind for `large' von Neumann algebras \(M\), i.e., algebras without a non-zero \(\sigma\)-finite direct summand: If \(X\) is a normed space with weak\(^*\) separable dual, then the ranges of \(M_1^+\) and \(P(M)\) coincide under a norm continuous linear map from \(M\) into \(X\). Theorems of this kind are also proved in various Jordan structures, such as JBW\(^*\) triples.A Beurling-Blecher-Labuschagne type theorem for Haagerup noncommutative \(L^p\) spaceshttps://zbmath.org/1472.460662021-11-25T18:46:10.358925Z"Bekjan, Turdebek N."https://zbmath.org/authors/?q=ai:bekjan.turdebek-n"Raikhan, Madi"https://zbmath.org/authors/?q=ai:raikhan.madiSummary: Let \(\mathcal{M}\) be a \(\sigma\)-finite von Neumann algebra, equipped with a normal faithful state \(\varphi\), and let \(\mathcal{A}\) be maximal subdiagonal subalgebra of \(\mathcal{M}\) and \(1\leq p<\infty\). We prove a Beurling-Blecher-Labuschagne type theorem for \(\mathcal{A}\)-invariant subspaces of Haagerup noncommutative \(L^p(\mathcal{A})\) and give a characterization of outer operators in Haagerup noncommutative \(H^p\)-spaces associated with \(\mathcal{A}\).Transition probability preserving maps on a Grassmann space in a semifinite factorhttps://zbmath.org/1472.460672021-11-25T18:46:10.358925Z"Gu, Weichen"https://zbmath.org/authors/?q=ai:gu.weichen"Wu, Wenming"https://zbmath.org/authors/?q=ai:wu.wenming"Yuan, Wei"https://zbmath.org/authors/?q=ai:yuan.weiSuppose that \(\mathcal{H}\) is a Hilbert space and \(\mathrm{Tr}\) is the canonical trace on \(B(\mathcal{H})\). Let \(\mathscr{P}_1\) be the Grassmann space of rank-one projections in \(B(\mathcal{H})\). Wigner's theorem states that if \(\varphi: \mathscr{P}_1 \to \mathscr{P}_1\) is a surjective map preserving the transition probability, i.e., \(\mathrm{Tr}(PQ) = \mathrm{Tr}(\varphi(P)\, \varphi(Q))\) for all \(P, Q \in \mathscr{P}_1\), then there is either a \(*\)-isomorphism or a \(*\)-anti-isomorphism \(\sigma: B(\mathcal{H}) \to B(\mathcal{H})\) such that \(\varphi(P) = \sigma(P)\) for every \(P \in \mathscr{P}_1\). The main aim of the paper under review is to generalize Wigner's theorem by providing a characterization of surjective transition probability preserving transformations on the Grassmann space of infinite projections in an infinite semifinite factor.An operad of non-commutative independences defined by treeshttps://zbmath.org/1472.460682021-11-25T18:46:10.358925Z"Jekel, David"https://zbmath.org/authors/?q=ai:jekel.david"Liu, Weihua"https://zbmath.org/authors/?q=ai:liu.weihuaSummary: %\DeclareMathSymbol{\boxright}{3}{mathb}{'151}
We study certain notions of \(N\)-ary non-commutative independence, which generalize free, Boolean, and monotone independence. For every rooted subtree \(\mathcal{T}\) of an \(N\)-regular rooted tree, we define the \(\mathcal{T}\)-free product of \(N\) non-commutative probability spaces and the \(\mathcal{T}\)-free additive convolution of \(N\) non-commutative laws.
These \(N\)-ary additive convolution operations form a topological symmetric operad which includes the free, Boolean, monotone, and anti-monotone convolutions, as well as the orthogonal and subordination convolutions. Using the operadic framework, the proof of convolution identities such as
%\(\mu\boxplus\nu=\mu\rhd ( \nu\boxempty\kern-9pt\vdash \mu ) \)
%\(\mu\boxplus\nu=\mu\rhd(\nu \boxright \mu)\)
\(\mu\boxplus\nu = \mu \vartriangleright (\nu\,\square\!\!\!\!\!\vdash \!\mu) \)
can be reduced to combinatorial manipulations of trees. In particular, we obtain a decomposition of the \(\mathcal{T} \)-free convolution into iterated Boolean and orthogonal convolutions, which generalizes work of \textit{R. Lenczewski} [J. Funct. Anal. 246, No. 2, 330--365 (2007; Zbl 1129.46055)].
We also develop a theory of \(\mathcal{T} \)-free independence that closely parallels the free, Boolean, and monotone cases, provided that the root vertex has more than one neighbor. This includes combinatorial moment formulas, cumulants, a central limit theorem, and classification of distributions that are infinitely divisible with bounded support. In particular, we study the case where the root vertex of \(\mathcal{T}\) has \(n\) children and each other vertex has \(d\) children, and we relate the \(\mathcal{T} \)-free convolution powers to free and Boolean convolution powers and the Belinschi-Nica semigroup.On the support of the free additive convolutionhttps://zbmath.org/1472.460692021-11-25T18:46:10.358925Z"Bao, Zhigang"https://zbmath.org/authors/?q=ai:bao.zhigang"Erdős, László"https://zbmath.org/authors/?q=ai:erdos.laszlo"Schnelli, Kevin"https://zbmath.org/authors/?q=ai:schnelli.kevinSummary: We consider the free additive convolution of two probability measures \(\mu\) and \(\nu\) on the real line and show that \(\mu\boxplus v\) is supported on a single interval if \(\mu\) and \(\nu\) each has single interval support. Moreover, the density of \(\mu\boxplus\nu\) is proven to vanish as a square root near the edges of its support if both \(\mu\) and \(\nu\) have power law behavior with exponents between \(-1\) and 1 near their edges. In particular, these results show the ubiquity of the conditions in our recent work on optimal local law at the spectral edges for addition of random matrices [\textit{Z.-G. Bao} et al., J. Funct. Anal. 279, No. 7, Article ID 108639, 93~p. (2020; Zbl 1460.46058)].Max-convolution semigroups and extreme values in limit theorems for the free multiplicative convolutionhttps://zbmath.org/1472.460702021-11-25T18:46:10.358925Z"Ueda, Yuki"https://zbmath.org/authors/?q=ai:ueda.yukiSummary: We investigate relations between additive convolution semigroups and max-con\-vo\-lu\-tion semigroups through the law of large numbers for the free multiplicative convolution. Based on these relations, we give a formula related with the Belinschi-Nica semigroup and the max-Belinschi-Nica semigroup. Finally, we give several limit theorems for classical, free and Boolean extreme values.Essential crossed products for inverse semigroup actions: simplicity and pure infinitenesshttps://zbmath.org/1472.460712021-11-25T18:46:10.358925Z"Kwaśniewski, Bartosz Kosma"https://zbmath.org/authors/?q=ai:kwasniewski.bartosz-kosma"Meyer, Ralf"https://zbmath.org/authors/?q=ai:meyer.ralf.1Summary: We study simplicity and pure infiniteness criteria for \(\mathrm{C}^*\)-algebras associated to inverse semigroup actions by Hilbert bimodules and to Fell bundles over étale not necessarily Hausdorff groupoids. Inspired by recent work of \textit{R. Exel} and \textit{D. R. Pitts} [``Characterizing groupoid \(\mathrm{C}^*\)-algebras of non-Hausdorff étale groupoids'', Preprint (2019); \url{arXiv: 1901.09683}], we introduce essential crossed products for which there are such criteria. In our approach the major role is played by a generalised expectation with values in the local multiplier algebra. We give a long list of equivalent conditions characterising when the essential and reduced \(\mathrm{C}^*\)-algebras coincide. Our most general simplicity and pure infiniteness criteria apply to aperiodic \(\mathrm{C}^*\)-inclusions equipped with supportive generalised expectations. We thoroughly discuss the relationship between aperiodicity, detection of ideals, purely outer inverse semigroup actions, and non-triviality conditions for dual groupoids.Categories of two-colored pair partitions. Part II: Categories indexed by semigroupshttps://zbmath.org/1472.460722021-11-25T18:46:10.358925Z"Mang, Alexander"https://zbmath.org/authors/?q=ai:mang.alexander"Weber, Moritz"https://zbmath.org/authors/?q=ai:weber.moritzSummary: Within the framework of unitary easy quantum groups, we study an analogue of Brauer's Schur-Weyl approach to the representation theory of the orthogonal group. We consider concrete combinatorial categories whose morphisms are formed by partitions of finite sets into disjoint subsets of cardinality two; the points of these sets are colored black or white. These categories correspond to ``half-liberated easy'' interpolations between the unitary group and Wang's quantum counterpart. We complete the classification of all such categories demonstrating that the subcategories of a certain natural halfway point are equivalent to additive subsemigroups of the natural numbers; the categories above this halfway point have been classified in a preceding article. We achieve this using combinatorial means exclusively. Our work reveals that the half-liberation procedure is quite different from what was previously known from the orthogonal case.
For Part I; see [\textit{A.~Mang} and \textit{M.~Weber}, Ramanujan J. 53, No.~1, 181--208 (2020; Zbl 07343721)].On the \(K\)-theory of \(C^*\)-algebras associated to substitution tilingshttps://zbmath.org/1472.460732021-11-25T18:46:10.358925Z"Gonçalves, Daniel"https://zbmath.org/authors/?q=ai:goncalves.daniel"Ramirez-Solano, Maria"https://zbmath.org/authors/?q=ai:ramirez-solano.mariaSummary: Under suitable conditions, a substitution tiling gives rise to a Smale space, from which three equivalence relations can be constructed, namely the stable, unstable, and asymptotic equivalence relations. We denote by \(S,U\), and \(A\) their corresponding \(C^*\)-algebras in the sense of Renault. We show that the \(K\)-theories of \(S\) and \(U\) can be computed from the cohomology and homology of a single cochain complex with connecting maps for tilings of the line and of the plane. Moreover, we provide formulas to compute the \(K\)-theory for these three \(C^*\)-algebras. Furthermore, we show that the \(K\)-theory groups for tilings of dimension~1 are always torsion free. For tilings of dimension~2, only \(K_0(U)\) and \(K_1(S)\) can contain torsion.Complex quantum groups and a deformation of the Baum-Connes assembly maphttps://zbmath.org/1472.460742021-11-25T18:46:10.358925Z"Monk, Andrew"https://zbmath.org/authors/?q=ai:monk.andrew-f"Voigt, Christian"https://zbmath.org/authors/?q=ai:voigt.christianSummary: We define and study an analogue of the Baum-Connes assembly map for complex semisimple quantum groups, that is, Drinfeld doubles of \(q\)-deformations of compact semisimple Lie groups. Our starting point is the deformation picture of the Baum-Connes assembly map for a complex semisimple Lie group \(G\), which allows one to express the \( K\)-theory of the reduced group \(C^\ast\)-algebra of \(G\) in terms of the \(K\)-theory of its associated Cartan motion group. The latter can be identified with the semidirect product of the maximal compact subgroup \(K\) acting on \( \mathfrak{k}^* \) via the coadjoint action.
In the quantum case the role of the Cartan motion group is played by the Drinfeld double of the classical group \(K\), whose associated group \( C^*\)-algebra is the crossed product of \( C(K) \) with respect to the adjoint action of \( K \). Our quantum assembly map is obtained by varying the deformation parameter in the Drinfeld double construction applied to the standard deformation \( K_q \) of \(K \). We prove that the quantum assembly map is an isomorphism, thus providing a description of the \( K \)-theory of complex quantum groups in terms of classical topology.
Moreover, we show that there is a continuous field of \( C^* \)-algebras which encodes both the quantum and classical assembly maps as well as a natural deformation between them. It follows in particular that the quantum assembly map contains the classical Baum-Connes assembly map as a direct summand.Noncommutative solenoidshttps://zbmath.org/1472.460752021-11-25T18:46:10.358925Z"Latrémolière, Frédéric"https://zbmath.org/authors/?q=ai:latremoliere.frederic"Packer, Judith"https://zbmath.org/authors/?q=ai:packer.judith-aAuthors' abstract: A noncommutative solenoid is a twisted group \(C^*\)-algebra \(C^*\left(\mathbb{Z}\left[\frac1{N}\right]^2,\sigma\right)\) where \(\mathbb{Z}\left[\frac1{N}\right]\) is the group of the \(N\)-adic rationals and \(\sigma\) is a multiplier of \(\mathbb{Z}\left[\frac1{N}\right]^2\). In this paper, we use techniques from noncommutative topology to classify these \(C^*\)-algebras up to \(*\)-isomorphism in terms of the multipliers of \(\mathbb{Z}\left[\frac1{N}\right]^2\). We also establish a necessary and sufficient condition for simplicity of noncommutative solenoids, compute their \(K\)-theory and show that the \(K_0\) groups of noncommutative solenoids are given by the extensions of \(\mathbb{Z}\) by \(\mathbb{Z}\left[\frac1{N}\right]\). We give a concrete description of non-simple noncommutative solenoids as bundle of matrices over solenoid groups, and we show that irrational noncommutative solenoids are real rank zero AT \(C^*\)-algebras.A weak expectation property for operator modules, injectivity and amenable actionshttps://zbmath.org/1472.460762021-11-25T18:46:10.358925Z"Bearden, Alex"https://zbmath.org/authors/?q=ai:bearden.alex"Crann, Jason"https://zbmath.org/authors/?q=ai:crann.jasonSolution analysis for a class of set-inclusive generalized equations: a convex analysis approachhttps://zbmath.org/1472.460772021-11-25T18:46:10.358925Z"Uderzo, Amos"https://zbmath.org/authors/?q=ai:uderzo.amosSummary: In the present paper, classical tools of convex analysis are used to study the solution set to a certain class of set-inclusive generalized equations. A condition for the solution existence and for global error bounds is established, in the case the set-valued term appearing in the generalized equation is concave. A functional characterization of the contingent (a.k.a. Bouligand tangent) cone to the solution set is provided via directional derivatives. Specializations of these results are also considered when outer prederivatives can be employed as approximations of set-valued mappings.A precision on the concept of strict convexity in non-Archimedean analysishttps://zbmath.org/1472.460782021-11-25T18:46:10.358925Z"Cabello Sánchez, Javier"https://zbmath.org/authors/?q=ai:cabello-sanchez.javier"Navarro Garmendia, José"https://zbmath.org/authors/?q=ai:navarro-garmendia.joseA non-Archimedean normed space \(X\) over a field \(K\) is called strictly convex if \(|2|= 1\) and for any pair of vectors \(x, y \in X\), \(\|x\|= \|y\|= \|x + y\|\) ensures that \(x = y\). \textit{A. Kubzdela} [Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods 75, No. 4, 2060--2068 (2012; Zbl 1248.46050)] showed that the only non-Archimedean strictly convex space over a field with a non-trivial valuation is the zero space. The authors prove that the only non-Archimedean strictly convex spaces are the zero space and the one-dimensional linear space over \(\mathbb{Z}/3\mathbb{Z}\) with any of its trivial norms.The Hahn-Banach extension theorem for fuzzy normed spaces revisitedhttps://zbmath.org/1472.460792021-11-25T18:46:10.358925Z"Alegre, Carmen"https://zbmath.org/authors/?q=ai:alegre.carmen"Romaguera, Salvador"https://zbmath.org/authors/?q=ai:romaguera.salvadorSummary: This paper deals with fuzzy normed spaces in the sense of \textit{S.-C. Cheng} and \textit{J. N. Mordeson} [Bull. Calcutta Math. Soc. 86, No. 5, 429--436 (1994; Zbl 0829.47063)]. We characterize fuzzy norms in terms of ascending and separating families of seminorms and prove an extension theorem for continuous linear functionals on a fuzzy normed space. Our result generalizes the classical Hahn-Banach extension theorem for normed spaces.The Bézout equation on the right half-plane in a Wiener space settinghttps://zbmath.org/1472.470122021-11-25T18:46:10.358925Z"Groenewald, G. J."https://zbmath.org/authors/?q=ai:groenewald.gilbert-j"ter Horst, S."https://zbmath.org/authors/?q=ai:ter-horst.sanne"Kaashoek, M. A."https://zbmath.org/authors/?q=ai:kaashoek.marinus-aSummary: This paper deals with the Bézout equation \({G}(s){X}(s) = {I}_{m}, \mathfrak{R}{s} \leq {0}\), in the Wiener space of analytic matrix-valued functions on the right halfplane. In particular, \(G\) is an \(m \times p\) matrix-valued analytic Wiener function, where \(p\geq m\), and the solution \(X\) is required to be an analytic Wiener function of size \(p\times m\). The set of all solutions is described explicitly in terms of a \(p\times p\) matrix-valued analytic Wiener function \(Y\), which has an inverse in the analytic Wiener space, and an associated inner function \(\Theta\) defined by \(Y\) and the value of \(G\) at infinity. Among the solutions, one is identified that minimizes the \(H^{2}\)-norm. A~Wiener space version of Tolokonnikov's lemma plays an important role in the proofs. The results presented are natural analogues of those obtained for the discrete case in [the authors, Complex Anal. Oper. Theory 10, No. 1, 115--139 (2016; Zbl 1337.47018)].
For the entire collection see [Zbl 1367.47005].Traces on operator ideals and related linear forms on sequence ideals. IVhttps://zbmath.org/1472.470162021-11-25T18:46:10.358925Z"Pietsch, Albrecht"https://zbmath.org/authors/?q=ai:pietsch.albrechtThe author continues in this paper a sequence of papers on the topic. Here, he deals with dyadic representations of a bounded linear operator \(S\in \mathcal L(X,Y)\), meaning that \(S=\sum_{k=0}^\infty S_k\) where \(S_k\in \mathcal L(X,Y)\) are finite rank operators with \(\operatorname{rank} (S_k)\leq 2^k\) and with shift-monotone sequences ideals \(\mathcal \xi(\mathbb N_0)\), that is, \(S_+\)-invariant linear subspaces of \(\ell_\infty(\mathbb N_0)\), where \(S_+\) stands for the forward shift, satisfying that, if \(a\in \mathcal \xi(\mathbb N_0)\), \(b\in \ell_\infty(\mathbb N_0)\) and \(\sup_{n\ge k}|b_n|\le \sup_{n\ge k}|a_n|\), then \(a\in \mathcal \xi(\mathbb N_0)\). The author refers to \(S=\sum_{k=0}^\infty S_k\) as an \((\mathcal U, \mathcal \xi)\)-representation if \((\|S-\sum_{k=0}^{n-1} S_k\|_{\mathcal U})_n\in \mathbb \xi(\mathbb N_0)\).
It is known [the author, Indag. Math., New Ser. 25, No. 2, 341--365 (2014; Zbl 1319.47067)] that there is a one-to-one correspondence between all symmetric sequence ideals and all shift-monotone sequence ideals. The correspondence maps to each shift-monotone sequence ideal \(\mathcal \xi (\mathbb N_0)\) the symmetric sequence ideal \(s(\mathbb N_0)=\{a\in \ell_\infty(\mathbb N_0): (a_{2^k})\in \mathcal \xi(\mathbb N_0)\}\) and the sequence ideals \(s(\mathbb N_0)\) and \(\xi(\mathbb N_0)\) are said to be associated.
The main result establishes that, if \(\mathcal U\) is a quasi Banach operator ideal of trace type \(\rho\), that is, \(|\operatorname{trace}(F)|\le c n^\rho\|F\|_{\mathcal U}\) whenever \(F\) is a finite rank operator with \(\operatorname{rank}(F)\le n\) and \(c\) is a constant independent of \(F\), and \(\mu\) is a \(2^{-\rho}S_{+}\)-invariant linear form on a shift-monotone sequence ideal \(\xi(\mathbb N_0)\), then \(\tau(S):=\mu(2^{-k\rho}\operatorname{trace}(S_k))\) does not depend on the choice of the \((\mathcal U, \xi)\)-representation \(S=\sum_{k=0}^\infty S_k\). This result is then applied to concrete examples of Banach operator ideals, such as those given by absolutely \((q,2)\)-summing operators and shift-monotone sequence ideals such as those associated to Lorentz sequence ideals. Concrete examples of operators such as convolution operators generated by functions in certain Lipschitz and Besov classes are provided.
For Parts I--III, see [the author, Indag. Math., New Ser. 25, No. 2, 341--365 (2014; Zbl 1319.47067); Integral Equations Oper. Theory 79, No. 2, 255--299 (2014; Zbl 1337.47031); J. Math. Anal. Appl. 421, No. 2, 971--981 (2015; Zbl 1328.47021)].
For the entire collection see [Zbl 1367.47005].On the norm of linear combinations of projections and some characterizations of Hilbert spaceshttps://zbmath.org/1472.470292021-11-25T18:46:10.358925Z"Krupnik, Nahum"https://zbmath.org/authors/?q=ai:krupnik.naum-yakovlevich"Markus, Alexander"https://zbmath.org/authors/?q=ai:markus.alexander-sSummary: Let \( \mathcal{B} \) be a Banach space and let \(P, Q\) (\(P, Q \neq 0\)) be two complementary projections in \( \mathcal{B}\) (i.e., \(P +Q=I\)). For dim \( \mathcal{B} > 2 \), we show that formulas of the kind \( \| aP \, +\, bQ \| = f(a, b, \| P\|) \) hold if and only if the norm in \( \mathcal{B} \) can be induced by an inner product. The two-dimensional case needs special consideration, which is done in the last two sections.
For the entire collection see [Zbl 1367.47005].The Kalton-Lancien theorem revisited: maximal regularity does not extrapolatehttps://zbmath.org/1472.470312021-11-25T18:46:10.358925Z"Fackler, Stephan"https://zbmath.org/authors/?q=ai:fackler.stephanSummary: We give a new more explicit proof of a result by \textit{N. J. Kalton} and \textit{G. Lancien} [Math. Z. 235, No. 3, 559--568 (2000; Zbl 1010.47024)] stating that on each Banach space with an unconditional basis not isomorphic to a Hilbert space there exists a generator $A$ of a holomorphic semigroup which does not have maximal regularity. In particular, we show that there always exists a Schauder basis \((f_m)\) such that $A$ can be chosen of the form \(A(\sum_{m = 1}^\infty a_m f_m) = \sum_{m = 1}^\infty 2^m a_m f_m\). Moreover, we show that maximal regularity does not extrapolate: we construct consistent holomorphic semigroups \((T_p(t))_{t \geqslant 0}\) on \(L^p(\mathbb{R})\) for \(p \in(1, \infty)\) which have maximal regularity if and only if \(p = 2\). These assertions were both open problems. Our approach is completely different than the one of Kalton and Lancien [loc.\,cit.]. We use the characterization of maximal regularity by \(\mathcal{R}\)-sectoriality for our construction.Wave front sets with respect to the iterates of an operator with constant coefficientshttps://zbmath.org/1472.470352021-11-25T18:46:10.358925Z"Boiti, C."https://zbmath.org/authors/?q=ai:boiti.chiara"Jornet, D."https://zbmath.org/authors/?q=ai:jornet.david"Juan-Huguet, J."https://zbmath.org/authors/?q=ai:juan-huguet.jordiSummary: We introduce the wave front set \(\text{WF}_*^P(u)\) with respect to the iterates of a hypoelliptic linear partial differential operator with constant coefficients of a classical distribution \(u \in \mathcal{D}'(\Omega)\) in an open set \(\Omega\) in the setting of ultradifferentiable classes of \textit{R. W. Braun} et al. [Result. Math. 17, No. 3--4, 206--237 (1990; Zbl 0735.46022)]. We state a version of the microlocal regularity theorem of Hörmander for this new type of wave front set and give some examples and applications of the former result.On monotone mappings in modular function spaceshttps://zbmath.org/1472.470402021-11-25T18:46:10.358925Z"Alfuraidan, M. R."https://zbmath.org/authors/?q=ai:alfuraidan.monther-rashed"Khamsi, M. A."https://zbmath.org/authors/?q=ai:khamsi.mohamed-amine"Kozlowski, W. M."https://zbmath.org/authors/?q=ai:kozlowski.wojciech-mSummary: Because of its many diverse applications, fixed point theory has been a flourishing area of mathematical research for decades. Banach's formulation of the contraction mapping principle in the early twentieth century signaled the advent of an intense interest in the metric related aspects of the theory. The metric fixed point theory in modular function spaces is closely related to the metric theory, in that it provides modular equivalents of norm and metric concepts. Modular spaces are extensions of the classical Lebesgue and Orlicz spaces, and in many instances, conditions cast in this framework are more natural and more easily verified than their metric analogs. In this chapter, we study the existence and construction of fixed points for monotone nonexpansive mappings acting in modular functions spaces equipped with a partial order or a graph structure.
For the entire collection see [Zbl 1470.47001].On a fixed point theorem in some nonlocally convex spaceshttps://zbmath.org/1472.470422021-11-25T18:46:10.358925Z"Bayoumi, Aboubakr"https://zbmath.org/authors/?q=ai:bayoumi.aboubakr"Ezat, Ibrahim"https://zbmath.org/authors/?q=ai:ezat.ibrahim(no abstract)Qualification conditions-free characterizations of the \(\varepsilon \)-subdifferential of convex integral functionshttps://zbmath.org/1472.490292021-11-25T18:46:10.358925Z"Correa, Rafael"https://zbmath.org/authors/?q=ai:correa.rafael"Hantoute, Abderrahim"https://zbmath.org/authors/?q=ai:hantoute.abderrahim"Pérez-Aros, Pedro"https://zbmath.org/authors/?q=ai:perez-aros.pedroSummary: We provide formulae for the \(\varepsilon \)-subdifferential of the integral function \(I_f(x):=\int_T f(t,x)\, d\mu (t)\), where the integrand \(f:T\times X \rightarrow \overline{\mathbb{R}}\) is measurable in \((t, x)\) and convex in \(x\). The state variable lies in a locally convex space, possibly non-separable, while \(T\) is given a structure of a nonnegative complete \(\sigma \)-finite measure space \((T,\mathcal{A},\mu )\). The resulting characterizations are given in terms of the \(\varepsilon \)-subdifferential of the data functions involved in the integrand, \(f\), without requiring any qualification conditions. We also derive new formulas when some usual continuity-type conditions are in force. These results are new even for the finite sum of convex functions and for the finite-dimensional setting.Geometric methods in physics XXXVIII. Workshop, Białowieża, Poland, June 30 -- July 6, 2019https://zbmath.org/1472.530062021-11-25T18:46:10.358925Z"Kielanowski, Piotr"https://zbmath.org/authors/?q=ai:kielanowski.piotr"Odzijewicz, Anatol"https://zbmath.org/authors/?q=ai:odzijewicz.anatol"Previato, Emma"https://zbmath.org/authors/?q=ai:previato.emmaPublisher's description: The book consists of articles based on the XXXVIII Białowieża Workshop on Geometric Methods in Physics, 2019. The series of Białowieża workshops, attended by a community of experts at the crossroads of mathematics and physics, is a major annual event in the field. The works in this book, based on presentations given at the workshop, are previously unpublished, at the cutting edge of current research, typically grounded in geometry and analysis, with applications to classical and quantum physics.
For the past eight years, the Białowieża Workshops have been complemented by a School on Geometry and Physics, comprising series of advanced lectures for graduate students and early-career researchers. The extended abstracts of the five lecture series that were given in the eighth school are included. The unique character of the Workshop-and-School series draws on the venue, a famous historical, cultural and environmental site in the Białowieża forest, a UNESCO World Heritage Centre in the east of Poland: lectures are given in the Nature and Forest Museum and local traditions are interwoven with the scientific activities.
The articles of this volume will be reviewed individually. For the preceding workshop see [Zbl 1433.53003].
Indexed articles:
\textit{Arici, Francesca; Mesland, Bram}, Toeplitz extensions in noncommutative topology and mathematical physics, 3-29 [Zbl 07400843]
\textit{Beltiţă, Daniel; Odzijewicz, Anatol}, Standard groupoids of von Neumann algebras, 31-39 [Zbl 07400844]
\textit{Cotti, Giordano}, Quantum differential equations and helices, 41-65 [Zbl 07400845]
\textit{Dobrogowska, Alina; Mironov, Andrey E.}, Periodic one-point rank one commuting difference operators, 67-74 [Zbl 1472.39037]
\textit{Fehér, L.; Marshall, I.}, On the bi-Hamiltonian structure of the trigonometric spin Ruijsenaars-Sutherland hierarchy, 75-87 [Zbl 07400847]
\textit{Hara, Kentaro}, Hermitian-Einstein metrics from non-commutative \(U(1)\) solutions, 89-96 [Zbl 07400848]
\textit{Hounkonnou, Mahouton Norbert; Houndédji, Gbêvèwou Damien}, 2-hom-associative bialgebras and hom-left symmetric dialgebras, 97-115 [Zbl 07400849]
\textit{Crus y Cruz, S.; Gress, Z.; Jiménez-Macías, P.; Rosas-Ortiz, O.}, Laguerre-Gaussian wave propagation in parabolic media, 117-128 [Zbl 07400850]
\textit{Karmanova, Maria}, Maximal surfaces on two-step sub-Lorentzian structures, 129-141 [Zbl 07400851]
\textit{Lawson, Jimmie D.; Lim, Yongdo}, Following the trail of the operator geometric mean, 143-153 [Zbl 07400852]
\textit{Mandal, Ashis; Mishra, Satyendra Kumar}, On Hom-Lie-Rinehart algebras, 155-161 [Zbl 07400853]
\textit{Nakayashiki, Atsushi}, One step degeneration of trigonal curves and mixing of solitons and quasi-periodic solutions of the KP equation, 163-186 [Zbl 07400854]
\textit{Dobrokhotov, Sergei; Nazaikinskii, Vladimir}, Fock quantization of canonical transformations and semiclassical asymptotics for degenerate problems, 187-195 [Zbl 1472.81099]
\textit{Nieto, L. M.; Gadella, M.; Mateos-Guilarte, J.; Muñoz-Castañeda, J. M.; Romaniega, C.}, Some recent results on contact or point supported potentials, 197-219 [Zbl 07400856]
\textit{Orlov, Aleksandr Yu.}, 2D Yang-Mills theory and tau functions, 221-250 [Zbl 07400857]
\textit{Prorok, Dominik; Prykarpatski, Anatolij}, Many-particle Schrödinger type finitely factorized quantum Hamiltonian systems and their integrability, 251-270 [Zbl 07400858]
\textit{Quintana, C.; Jiménez-Macías, P.; Rosas-Ortiz, O.}, Quantum master equation for the time-periodic density operator of a single qubit coupled to a harmonic oscillator, 271-281 [Zbl 1472.81131]
\textit{Zelaya, Kevin; Cruz y Cruz, Sara; Rosas-Ortiz, Oscar}, On the construction of non-Hermitian Hamiltonians with all-real spectra through supersymmetric algorithms, 283-292 [Zbl 1472.81109]
\textit{Sontz, Stephen Bruce}, Toeplitz quantization of an analogue of the Manin plane, 293-301 [Zbl 07400861]
\textit{Przanowski, Maciej; Tosiek, Jaromir; Turrubiates, Francisco J.}, The Weyl-Wigner-Moyal formalism on a discrete phase space, 303-312 [Zbl 1472.81132]
\textit{Zheglov, Alexander}, Algebraic geometric properties of spectral surfaces of quantum integrable systems and their isospectral deformations, 313-331 [Zbl 1470.13043]
\textit{Domrin, A. V.}, Soliton equations and their holomorphic solutions, 335-343 [Zbl 07400864]
\textit{Goldin, Gerald A.}, Diffeomorphism groups in quantum theory and statistical physics, 345-350 [Zbl 1472.81114]
\textit{Rosas-Ortiz, Oscar}, Position-dependent mass systems: classical and quantum pictures, 351-361 [Zbl 07400866]
\textit{Slavnov, N. A.}, Introduction to the algebraic Bethe ansatz, 363-371 [Zbl 07400867]
\textit{Szymański, Wojciech}, Noncommutative fiber bundles, 373-379 [Zbl 07400868]A nonlinear Korn inequality on a surface with an explicit estimate of the constanthttps://zbmath.org/1472.530102021-11-25T18:46:10.358925Z"Malin, Maria"https://zbmath.org/authors/?q=ai:malin.maria"Mardare, Cristinel"https://zbmath.org/authors/?q=ai:mardare.cristinelSummary: A nonlinear Korn inequality on a surface estimates a distance between a surface \(\theta(\omega)\) and another surface \(\phi(\omega)\) in terms of distances between their fundamental forms in the space \(L^p(\omega)\), \(1<p<\infty\).
We establish a new inequality of this type. The novelty is that the immersion \(\theta\) belongs to a specific set of mappings of class \(\mathcal{C}^1\) from \(\overline{\omega}\) into \(\mathbb{R}^3\) with a unit vector field also of class \(\mathcal{C}^1\) over \(\overline{\omega}\).The \(L^p\)-Calderón-Zygmund inequality on non-compact manifolds of positive curvaturehttps://zbmath.org/1472.530512021-11-25T18:46:10.358925Z"Marini, Ludovico"https://zbmath.org/authors/?q=ai:marini.ludovico"Veronelli, Giona"https://zbmath.org/authors/?q=ai:veronelli.gionaSummary: We construct, for \(p> n\), a concrete example of a complete non-compact \(n\)-dimensional Riemannian manifold of positive sectional curvature which does not support any \(L^p\)-Calderón-Zygmund inequality:
\[
\begin{aligned}\Vert\mathrm{Hess}\varphi\Vert_{L^p}\le C(\Vert\varphi\Vert_{L^p}+\Vert\Delta\varphi\Vert_{L^p}),\qquad\forall\varphi\in C^{\infty}_c(M).\end{aligned}
\]
The proof proceeds by local deformations of an initial metric which (locally) Gromov-Hausdorff converge to an Alexandrov space. In particular, we develop on some recent interesting ideas by De Philippis and Núñez-Zimbron dealing with the case of compact manifolds. As a straightforward consequence, we obtain that the \(L^p\)-gradient estimates and the \(L^p\)-Calderón-Zygmund inequalities are generally not equivalent, thus answering an open question in the literature. Finally, our example gives also a contribution to the study of the (non-)equivalence of different definitions of Sobolev spaces on manifolds.Correction to: ``Distinguished \(C_p(X)\) spaces''https://zbmath.org/1472.540042021-11-25T18:46:10.358925Z"Ferrando, J. C."https://zbmath.org/authors/?q=ai:ferrando.juan-carlos"Kąkol, J."https://zbmath.org/authors/?q=ai:kakol.jerzy"Leiderman, A."https://zbmath.org/authors/?q=ai:leiderman.arkady-g"Saxon, S. A."https://zbmath.org/authors/?q=ai:saxon.stephen-aCorrection to the authors' paper [ibid. 115, No. 1, Paper No. 27, 18 p. (2021; Zbl 1460.54011)].The space consisting of uniformly continuous functions on a metric measure space with the \(L^p\) normhttps://zbmath.org/1472.540062021-11-25T18:46:10.358925Z"Koshino, Katsuhisa"https://zbmath.org/authors/?q=ai:koshino.katsuhisaLet \(\mathbf{s} = (-1,1)^\mathbb{N}\) be a countable infinite product of lines endowed with the product topology and let \(c_0\) be the subspace of \(\mathbf{s}\) consisting of those sequences converging to zero. Kadec, Bessaga, Pelczynski and other well-known mathematicians studied homeomorphisms between various infinite dimensional Banach and Fréchet spaces motivated by several questions posed by Fréchet and Banach. A classical celebrated result due to \textit{R. D. Anderson} [Bull. Am. Math. Soc. 72, 515--519 (1966; Zbl 0137.09703)] and \textit{M. I. Kadets} [Funkts. Anal. Prilozh. 1, No. 1, 61--70 (1967; Zbl 0166.10603)] states that every separable infinite dimensional Banach space or Fréchet space is homeomorphic to \(\mathbf{s}\).
In contrast with this result, \textit{R. Cauty} showed in [Fundam. Math. 139, No. 1, 23--36 (1991; Zbl 0793.46008)] that the subspace of \(L^p[0,1]\) consisting of those elements having a representative which is continuous is homeomorphic to \(c_0\) for any \(1\leq p < \infty\). In this paper the author generalizes the aforementioned result of Cauty showing that if \(X\) is a metric measure space satisfying some natural conditions then the subspace \(C_u(X)\) of \(L^p(X)\) consisting of those elements having a representative which is uniformly continuous is homeomorphic to \(c_0\) for any \(1\leq p < \infty\).Intermediate rings of complex-valued continuous functionshttps://zbmath.org/1472.540072021-11-25T18:46:10.358925Z"Acharyya, Amrita"https://zbmath.org/authors/?q=ai:acharyya.amrita"Acharyya, Sudip Kumar"https://zbmath.org/authors/?q=ai:acharyya.sudip-kumar"Bag, Sagarmoy"https://zbmath.org/authors/?q=ai:bag.sagarmoy"Sack, Joshua"https://zbmath.org/authors/?q=ai:sack.joshuaThe ring of complex-valued continuous functions on \(X\), where \(X\) is a completely regular Hausdorff topological space, is denoted by \(C(X, \mathbb{C})\). \(C^*(X, \mathbb{C})\) is its subring of bounded functions, \(\Sigma (X, C)\) is the collection of rings lying between \(C^*(X, \mathbb{C})\) and \(C(X, \mathbb{C})\). This paper shows extensive complex analogues of parallel results on absolutely convex, prime, maximal, \(z\)-, \(z^0\)-ideals of the intermediate rings of real-valued continuous functions on \(X\). Using a complex analogue of the structure space on the set of all maximal ideals on a commutative ring with unity (where a family of sets of maximal ideals forms a base for closed sets of the hull-kernel topology), the authors show that the complex analogue of the structure space of an intermediate ring \(P(X, \mathbb{C}) \in \Sigma(X, C)\) is the \textit{Stone-Čech} compactification \(\beta X\) of \(X\).
Extending the notion of real-valued \(C\)-type intermediate rings to rings of complex-valued continuous functions, \(P(X, \mathbb{C})\) is a \(C\)-type ring if it is isomorphic to a ring \(C(Y, \mathbb{C})\) for some Tychonoff space \(Y\). The ring \(C^*(X, \mathbb{C}) + I\), where \(I\) is a \(z\)-ideal in \(C(X, \mathbb{C})\), is a \(C\)-type intermediate ring of \(C(X, \mathbb{C})\). Those are the only \(C\)-type rings between \(C^*(X, \mathbb{C})\) and \(C(X, \mathbb{C})\) if and only if \(X\) is pseudocompact. The paper shows that for any maximal ideal \(M\) in \(C(X)\) and its complex analogue \(M_c\), the residue class field \(C(X, \mathbb{C} )/M_c\) is an algebraically closed field, as well as the algebraic closure of \(C(X)/M\). Some special cases are examined.
Ideals \(C_\mathcal{P}(X)\), where \(\mathcal{P}\) is an ideal of closed sets in \(X\), were introduced in [\textit{S. K. Acharyya} and \textit{S. K. Ghosh}, Topol. Proc. 35, 127--148 (2010; Zbl 1180.54040)] and investigated in [ibid. 40, 297--301 (2012; Zbl 1266.54057); \textit{S. Bag} et al., Appl. Gen. Topol. 20, No. 1, 109--117 (2019; Zbl 1429.54024)]. Here, a necessary and sufficient condition is found for the complex analogue \(C_\mathcal{P}(X, \mathbb{C})\) of \(C_\mathcal{P}(X)\) consisting of all functions with support (closure of the set of points where the functions are non-zero) in \(\mathcal{P}\) to be a prime ideal in \(C(X, \mathbb{C})\).
Also found are some estimates for certain parameters for zero divisor graphs [\textit{F. Azarpanah} and \textit{M. Motamedi}, Acta Math. Hung. 108, No. 1--2, 25--36 (2005; Zbl 1092.54007)] of an intermediate ring in \(\Sigma (X, C)\).On the cardinality of non-isomorphic intermediate rings of \(C(X)\)https://zbmath.org/1472.540082021-11-25T18:46:10.358925Z"Bose, B."https://zbmath.org/authors/?q=ai:bose.benjamin|bose.bedanta|bose.bella"Acharyya, S. K."https://zbmath.org/authors/?q=ai:acharyya.sudip-kumar|acharyya.s-kThe authors study ``intermediate rings'', which are subrings of the ring of real valued continuous functions which contains the ring of bounded real valued continuous functions.
To each intermediate ring \(A(X)\), it is associated a subspace \(\nu_A(X)\) of the Stone-Čech compactification of \(X\), which is an analogue of the Hewitt realcompactification. Also, they consider \([A(X)]\), the class of intermediate rings with homeomorphic subspaces \(\nu_A(X)\).
Some results in this context are obtained, for example: for a locally compact, non-compact but realcompact space \(X\), each class \([A(X)]\) has cardinality greater than \(2^c\); for a first countable noncompact realcompact space \(X\), there exist at least \(2^c\) intermediate subrings of \([A(X)]\), no two of which are isomorphic.Abundance of isomorphic and non isomorphic \(C\)-type intermediate ringshttps://zbmath.org/1472.540092021-11-25T18:46:10.358925Z"Bose, Bedanta"https://zbmath.org/authors/?q=ai:bose.bedanta"Acharyya, Sudip Kumar"https://zbmath.org/authors/?q=ai:acharyya.sudip-kumarAuthors' abstract: For a nonpseudocompact space \(X\), the family \(\Sigma (X)\) of all intermediate subrings of \(C(X)\) which contain \(C^*(X)\) contains at least \(2^c\) many distinct rings. We show that if in addition \(X\) is first countable and realcompact, then there are at least \(2^c\) many \(C\)-type intermediate rings in \(\Sigma (X)\) no two of which are pairwise isomorphic. With the special case \(X = \mathbb{N}\), it is shown that there exists a family containing \(c\)-many pairwise isomorphic \(C\)-type intermediate rings in \(\Sigma (\mathbb{N})\).On the sum of \(z^\circ\)-ideals in two classes of subrings of \(C(X)\)https://zbmath.org/1472.540102021-11-25T18:46:10.358925Z"Dube, Themba"https://zbmath.org/authors/?q=ai:dube.themba"Parsinia, Mehdi"https://zbmath.org/authors/?q=ai:parsinia.mehdiLet \(X\) be a Tychonoff space, \(C(X)\) be the ring of real-valued continuous functions on \(X\), \(I\) be an ideal of a subring \(A(X)\) of \(C(X)\) and \(C^*(X)\subseteq A(X)\). Then \(I\) is called a \(z^\circ\)-ideal if for every \(a \in I,~ P_a \subseteq I\), where \(P_a\) denotes the intersection of all minimal prime ideals of \(A(X)\) that contain \(a\). Generally, the goal of this paper is to consider sums of \(z^\circ\)-ideals in intermediate rings of \(C(X)\). Recall that \(X\) is called an \(F\)-space if every cozero-set in \(X\) is \(C^*\)-embedded. Algebraically, \(X\) is an \(F\)-space if and only if every finitely generated ideal of \(C(X)\) is principal. Recall that if every dense cozero-set is \(C^*\)-embedded in \(X\), then \(X\) is called a quasi \(F\)-space.
In this paper it is proved that \(X\) is a quasi \(F\)-space if and only if the sum of any two \(z^\circ\)-ideals in \(A(X)\) (any intermediate ring) is a \(z^\circ\)-ideal or all of \(A(X)\) (the whole intermediate ring). Let \(X\) be an almost \(P\)-space. Then, \(X\) is an \(F\)-space if and only if for each ideal \(I\) in \(C(X)\), such that \(I+R\) generates the topology on \(X\), the sum of any two \(z^\circ\)-ideals in \(I+\mathbb{R}\) is a \(z^\circ\)-ideal in \(I+\mathbb{R}\) or all of \(I+\mathbb{R}\).Some unitary representations of Thompson's groups \(F\) and \(T\)https://zbmath.org/1472.570142021-11-25T18:46:10.358925Z"Jones, Vaughan F. R."https://zbmath.org/authors/?q=ai:jones.vaughan-f-rSummary: In a ``naive'' attempt to create algebraic quantum field theories on the circle, we obtain a family of unitary representations of Thompson's groups \(T\) and \(F\) for any subfactor. The Thompson group elements are the ``local scale transformations'' of the theory. In a simple case the coefficients of the representations are polynomial invariants of links. We show that all links arise and introduce new ``oriented'' subgroups of \(\overrightarrow{F} < F\) and \(\overrightarrow{T} < T\) which allow us to produce all oriented knots and links.Recent advances in \(L^p\)-theory of homotopy operator on differential formshttps://zbmath.org/1472.580012021-11-25T18:46:10.358925Z"Ding, Shusen"https://zbmath.org/authors/?q=ai:ding.shusen"Shi, Peilin"https://zbmath.org/authors/?q=ai:shi.peilin"Wang, Yong"https://zbmath.org/authors/?q=ai:wang.yong.7|wang.yong.2|wang.yong.6|wang.yong.10|wang.yong|wang.yong.8|wang.yong.3|wang.yong.1|wang.yong.9|wang.yong.5Summary: The purpose of this survey paper is to present an up-to-date account of the recent advances made in the study of \(L^p\)-theory of the homotopy operator applied to differential forms. Specifically, we will discuss various local and global norm estimates for the homotopy operator \(T\) and its compositions with other operators, such as Green's operator and potential operator.A Kastler-Kalau-Walze type theorem for 7-dimensional manifolds with boundaryhttps://zbmath.org/1472.580192021-11-25T18:46:10.358925Z"Wang, Jian"https://zbmath.org/authors/?q=ai:wang.jian.3"Wang, Yong"https://zbmath.org/authors/?q=ai:wang.yongSummary: We give a brute-force proof of the Kastler-Kalau-Walze type theorem for 7-dimensional manifolds with boundary.On decoupling in Banach spaceshttps://zbmath.org/1472.600352021-11-25T18:46:10.358925Z"Cox, Sonja"https://zbmath.org/authors/?q=ai:cox.sonja-gisela"Geiss, Stefan"https://zbmath.org/authors/?q=ai:geiss.stefanSummary: We consider decoupling inequalities for random variables taking values in a Banach space \(X\). We restrict the class of distributions that appear as conditional distributions while decoupling and show that each adapted process can be approximated by a Haar-type expansion in which only the pre-specified conditional distributions appear. Moreover, we show that in our framework a progressive enlargement of the underlying filtration does not affect the decoupling properties (in particular, it does not affect the constants involved). As a special case, we deal with one-sided moment inequalities for decoupled dyadic (i.e., Paley-Walsh) martingales and show that Burkholder-Davis-Gundy-type inequalities for stochastic integrals of \(X\)-valued processes can be obtained from decoupling inequalities for \(X\)-valued dyadic martingales.The entrance law of the excursion measure of the reflected process for some classes of Lévy processeshttps://zbmath.org/1472.600822021-11-25T18:46:10.358925Z"Chaumont, Loïc"https://zbmath.org/authors/?q=ai:chaumont.loic"Małecki, Jacek"https://zbmath.org/authors/?q=ai:malecki.jacekSummary: We provide integral formulae for the Laplace transform of the entrance law of the reflected excursions for symmetric Lévy processes in terms of their characteristic exponent. For subordinate Brownian motions and stable processes we express the density of the entrance law in terms of the generalized eigenfunctions for the semigroup of the process killed when exiting the positive half-line. We use the formulae to study in-depth properties of the density of the entrance law such as asymptotic behavior of its derivatives in time variable.Spectral gap in mean-field \({\mathcal{O}}(n)\)-modelhttps://zbmath.org/1472.601502021-11-25T18:46:10.358925Z"Becker, Simon"https://zbmath.org/authors/?q=ai:becker.simon"Menegaki, Angeliki"https://zbmath.org/authors/?q=ai:menegaki.angelikiSummary: We study the dependence of the spectral gap for the generator of the Ginzburg-Landau dynamics for all \(\mathcal O(n)\)-\textit{models} with mean-field interaction and magnetic field, below and at the critical temperature on the number \(N\) of particles. For our analysis of the Gibbs measure, we use a one-step renormalization approach and semiclassical methods to study the eigenvalue-spacing of an auxiliary Schrödinger operator.Semi-parametric adjustment to computer modelshttps://zbmath.org/1472.620402021-11-25T18:46:10.358925Z"Wang, Yan"https://zbmath.org/authors/?q=ai:wang.yan.4|wang.yan.6|wang.yan.1|wang.yan.2|wang.yan|wang.yan.5|wang.yan.3"Tuo, Rui"https://zbmath.org/authors/?q=ai:tuo.ruiSummary: Computer simulations are widely used in scientific exploration and engineering designs. However, computer outputs usually do not match the reality perfectly because the computer models are built under certain simplifications and approximations. When physical observations are also available, statistical methods can be applied to estimate the discrepancy between the computer output and the physical response. In this article, we propose a semi-parametric method for statistical adjustments to computer models. The proposed method is proven to enjoy nice theoretical properties. We use three numerical studies and a real example to examine the predictive performance of the proposed method. The results show that the proposed method outperforms existing methods.Optimal prediction for high-dimensional functional quantile regression in reproducing kernel Hilbert spaceshttps://zbmath.org/1472.621772021-11-25T18:46:10.358925Z"Yang, Guangren"https://zbmath.org/authors/?q=ai:yang.guangren"Liu, Xiaohui"https://zbmath.org/authors/?q=ai:liu.xiaohui"Lian, Heng"https://zbmath.org/authors/?q=ai:lian.hengSummary: Regression problems with multiple functional predictors have been studied previously. In this paper, we investigate functional quantile linear regression with multiple functional predictors within the framework of reproducing kernel Hilbert spaces. The estimation procedure is based on an \(\ell_1\)-mixed-norm penalty. The learning rate of the estimator in prediction loss is established and a lower bound on the learning rate is also presented that matches the upper bound up to a logarithmic term.Covariant CP-instruments and their convolution semigroupshttps://zbmath.org/1472.810152021-11-25T18:46:10.358925Z"Heo, Jaeseong"https://zbmath.org/authors/?q=ai:heo.jaeseong"Ji, Un Cig"https://zbmath.org/authors/?q=ai:ji.un-cigSummary: Using probability operators and Fourier transforms of CP-instruments on von Neumann algebras, we give necessary and sufficient conditions for operators to be probability operators associated with covariant CP-instruments or to be Fourier transforms of covariant CP-instruments. We discuss a convolution semigroup of covariant CP-instruments and a semigroup of probability operators associated with CP-instruments on von Neumann algebras.Improving direct state measurements by using rebits in real enlarged Hilbert spaceshttps://zbmath.org/1472.810162021-11-25T18:46:10.358925Z"Ho, Le Bin"https://zbmath.org/authors/?q=ai:ho.le-binSummary: We propose a protocol to improve the accuracy of direct complex state measurements (DSM) by using rebits in real Hilbert spaces. We show that to improve the accuracy, the initial complex state should be decomposed into the real and imaginary parts and stored in an extended state (rebit) which can be tracked individually by two bases of an extra qubit. For pure states, the numerical calculations show that the trace distances between the true state and the reconstructed state obtained from the rebit method are more precise than those ones obtained from the usual DSM and quantum state tomography (SQT) because the number of projective measurements is reduced. For mixed states, the rebit method gives the same accuracy in comparison to the usual DSM, while it is less precise than QST. Its precision is also significantly improved when using nearly-pure states. Our proposal holds promises as a reliable tool for quantum computation, testing of quantum circuits by using only real amplitudes.The influence of non-Gaussian noise on weak valueshttps://zbmath.org/1472.810172021-11-25T18:46:10.358925Z"Ma, Fang-Yuan"https://zbmath.org/authors/?q=ai:ma.fang-yuan"Li, Jun-Gang"https://zbmath.org/authors/?q=ai:li.jun-gang"Zou, Jian"https://zbmath.org/authors/?q=ai:zou.jianSummary: The influence of Gaussian noise on weak values is studied. A general expression of weak values is derived, which is applicable to both Markovian and non-Markovian environment. Weak values under random telegraph noise and the colored noise of type \(1/f^a\) are investigated in particular, and the properties of weak values under those non-Gaussian noises are discussed. Furthermore, the threshold time for weak values keeping its characteristic to exceed spectrum range is found, which can reach a large value by the memory effect of non-Markovian environment.A steady state quantum classifierhttps://zbmath.org/1472.810192021-11-25T18:46:10.358925Z"Türkpençe, Deniz"https://zbmath.org/authors/?q=ai:turkpence.deniz"Akıncı, Tahir Çetin"https://zbmath.org/authors/?q=ai:akinci.tahir-cetin"Şeker, Serhat"https://zbmath.org/authors/?q=ai:seker.serhatSummary: We report that under some specific conditions a single qubit model weakly interacting with information environments can be referred to as a quantum classifier. We exploit the additivity and the divisibility properties of the completely positive (CP) quantum dynamical maps in order to obtain an open quantum classifier. The steady state response of the system with respect to the input parameters was numerically investigated and it's found that the response of the open quantum dynamics at steady state acts non-linearly with respect to the input data parameters. We also demonstrate the linear separation of the quantum data instances that reflects the success of the functionality of the proposed model both for ideal and experimental conditions. Superconducting circuits were pointed out as the physical model to implement the theoretical model with possible imperfections.Quantum information measures of the Aharonov-Bohm ring in uniform magnetic fieldshttps://zbmath.org/1472.810232021-11-25T18:46:10.358925Z"Olendski, O."https://zbmath.org/authors/?q=ai:olendski.olegSummary: Shannon quantum information entropies \(S_{\rho, \gamma}\), Fisher informations \(I_{\rho, \gamma}\), Onicescu energies \(O_{\rho, \gamma}\) and complexities \(e^S O\) are calculated both in the position (subscript \(\rho\)) and momentum (\(\gamma\)) spaces for the azimuthally symmetric two-dimensional nanoring that is placed into the combination of the transverse uniform magnetic field \(\mathbf{B}\) and the Aharonov-Bohm (AB) flux \(\phi_{AB}\) and whose potential profile is modelled by the superposition of the quadratic and inverse quadratic dependencies on the radius \(r\). The increasing intensity \(B\) flattens momentum waveforms \(\Phi_{nm}(\mathbf{k})\) and in the limit of the infinitely large fields they turn to zero: \(\Phi_{n m}(\mathbf{k}) \to 0\) at \(B \to \infty \), what means that the position wave functions \(\Psi_{n m}(\mathbf{r})\), which are their Fourier counterparts, tend in this limit to the \(\delta\)-functions. Position (momentum) Shannon entropy depends on the field \(B\) as a negative (positive) logarithm of \(\omega_{eff} \equiv (\omega_0^2 + \omega_c^2 / 4)^{1/2}\), where \(\omega_0\) determines the quadratic steepness of the confining potential and \(\omega_c\) is a cyclotron frequency. This makes the sum \(S_{\rho_{nm}} + S_{\gamma_{nm}}\) a field-independent quantity that increases with the principal \(n\) and azimuthal \(m\) quantum numbers and does satisfy entropic uncertainty relation. Position Fisher information does not depend on \(m\), linearly increases with \(n\) and varies as \(\omega_{eff}\) whereas its \(n\) and \(m\) dependent Onicescu counterpart \(O_{\rho_{nm}}\) changes as \(\omega_{eff}^{-1}\). The products \(I_{\rho_{nm}} I_{\gamma_{nm}}\) and \(O_{\rho_{nm}} O_{\gamma_{nm}}\) are \(B\)-independent quantities. A dependence of the measures on the ring geometry is discussed. It is argued that a variation of the position Shannon entropy or Onicescu energy with the AB field uniquely determines an associated persistent current as a function of \(\phi_{AB}\) at \(B = 0\). An inverse statement is correct too.Entanglement breaking channels, stochastic matrices, and primitivityhttps://zbmath.org/1472.810252021-11-25T18:46:10.358925Z"Ahiable, Jennifer"https://zbmath.org/authors/?q=ai:ahiable.jennifer"Kribs, David W."https://zbmath.org/authors/?q=ai:kribs.david-w"Levick, Jeremy"https://zbmath.org/authors/?q=ai:levick.jeremy"Pereira, Rajesh"https://zbmath.org/authors/?q=ai:pereira.rajesh"Rahaman, Mizanur"https://zbmath.org/authors/?q=ai:rahaman.mizanurSummary: We consider the important class of quantum operations (completely positive trace-preserving maps) called entanglement breaking channels. We show how every such channel induces stochastic matrix representations that have the same non-zero spectrum as the channel. We then use this to investigate when entanglement breaking channels are primitive, and prove this depends on primitivity of the matrix representations. This in turn leads to tight bounds on the primitivity index of entanglement breaking channels in terms of the primitivity index of the associated stochastic matrices. We also present examples and discuss open problems generated by the work.Extremal states of qubit-qutrit system with maximally mixed marginalshttps://zbmath.org/1472.810292021-11-25T18:46:10.358925Z"Kanmani, S."https://zbmath.org/authors/?q=ai:kanmani.s-s"Satyanarayana, S. V. M."https://zbmath.org/authors/?q=ai:satyanarayana.s-v-mSummary: We study extremal elements of the convex set of qubit-qutrit states whose marginals are maximally mixed. In the two qubit case, it is known that every extreme state of such a convex set is a maximally entangled pure state. In qubit-qutrit case, pure states do not exist in the convex set. We construct mixed extreme states of ranks 2 and 3. Second rank extremal state is entangled whereas third rank extreme element is separable. Parthasarathy obtained an upper bound on the rank of extreme states of such a convex set of a bipartite system of \(n\) and \(m\) dimensions as \(\sqrt{ n^2+m^2-1}\). Thus for a qubit-qutrit system, the rank of an extreme element should be less than \(\sqrt{12} \). Since Parthasarathy's bound for two qubit system is \(\sqrt{7}\) and all extreme elements are of rank one, Rudolph posed a question about its tightness. We establish that Parthasarathy's upper bound is tight for qubit-qutrit system.Strong entanglement criterion involving momentum weak valueshttps://zbmath.org/1472.810362021-11-25T18:46:10.358925Z"Valdés-Hernández, A."https://zbmath.org/authors/?q=ai:valdes-hernandez.andrea"de la Peña, L."https://zbmath.org/authors/?q=ai:pena.l-de-la"Cetto, A. M."https://zbmath.org/authors/?q=ai:cetto.ana-mariaSummary: In recent years weak values have been used to explore interesting quantum features in novel ways. In particular, the real part of the weak value of the momentum operator has been widely studied, mainly in connection with Bohmian trajectories. Here we focus on the imaginary part and its role in relation with the entanglement of a bipartite system. We establish an entanglement criterion based on weak momentum correlations, that allows to discern whether the entanglement is encoded in the amplitude and/or in the phase of the wave function. Our results throw light on the physical role of the real and imaginary parts of the weak values, and stress the relevance of the latter in the multi-particle scenario.Tosio Kato's work on non-relativistic quantum mechanics. IIhttps://zbmath.org/1472.810942021-11-25T18:46:10.358925Z"Simon, Barry"https://zbmath.org/authors/?q=ai:simon.barry.1The work is the second part of a review to Kato's work on nonrelativistic quantum mechanics. It focuses on bounds on the number of eigenvalues of the helium atom, on the absence of embedded bound states, on scattering theory under a trace class condition, Kato smoothness, the adiabatic theorem, and the Trotter product formula.
The author is known for the clarity of his presentation which is reflected in this work as well. The review can also serve as an introduction of the subject, since the results are not merely reviewed but put in a current perspective of the field. An example of this is the appendix where the inequality \(|p|>2/(\pi |x|)\) in \(d=3\), known as Kato's inequality or Herbst inequality, is treated. It is put in the context of the groundstate transform which [\textit{R. L. Frank} et al., J. Am. Math. Soc. 21, No. 4, 925--950 (2008; Zbl 1202.35146)] used to prove a generalization for fractional powers of \(p:=-i\nabla\).
For Part I see the author [Bull. Math. Sci. 8, No. 1, 121--232 (2018; Zbl 1416.81063)]Supertime and Pauli's principlehttps://zbmath.org/1472.811082021-11-25T18:46:10.358925Z"Musin, Y. R."https://zbmath.org/authors/?q=ai:musin.yu-rSummary: Connection of Pauli's principle with the nontrivial structure of the fermion supertime is shown. When supersymmetry is localized as supergravitation, fields of gravitational and exchange interaction carriers arise. The exchange interaction quantum of free fermions, being a superpartner of graviton (gravitino), is interpreted as a paulino -- the particle responsible for the effect of mutual avoidance of identical fermions.Characterization of dynamical measurement's uncertainty in a two-qubit system coupled with bosonic reservoirshttps://zbmath.org/1472.811262021-11-25T18:46:10.358925Z"Chen, Min-Nan"https://zbmath.org/authors/?q=ai:chen.min-nan"Wang, Dong"https://zbmath.org/authors/?q=ai:wang.dong.3"Ye, Liu"https://zbmath.org/authors/?q=ai:ye.liuSummary: We study the dynamical characteristics of the entropy-based uncertainty with regard to a pair of incompatible measurements under a bipartite qubit-system suffering from quantum decoherence induced by hierarchical environments. How non-Markovian and Markovian environments affect the dynamical behaviors of the measurement's uncertainty is revealed. We prove that the measured uncertainty of interest demonstrates a non-monotonic behavior, viz., the amount will increase initially and subsequently oscillate periodically with the growth of time in a non-Markovian regime; On the contrary, the uncertainty will inflate firstly and monotonically decrease in a Markovian regime. Noteworthily, we put forward a simple and feasible strategy to suppress the damping of the system and hence be good for decreasing the magnitude of the uncertainty, by virtue of optimal combination of pre-weak measurements and post-filtering operations. Furthermore, we explore the applications of the uncertainty relation investigated on entanglement witness and channel capacity.The renormalization structure of \(6D\), \(\mathcal{N} = (1, 0)\) supersymmetric higher-derivative gauge theoryhttps://zbmath.org/1472.811612021-11-25T18:46:10.358925Z"Buchbinder, I. L."https://zbmath.org/authors/?q=ai:buchbinder.ioseph-l"Ivanov, E. A."https://zbmath.org/authors/?q=ai:ivanov.evgenii-alekseevich"Merzlikin, B. S."https://zbmath.org/authors/?q=ai:merzlikin.boris-s"Stepanyantz, K. V."https://zbmath.org/authors/?q=ai:stepanyantz.k-vSummary: We consider the harmonic superspace formulation of higher-derivative \(6D\), \(\mathcal{N} = (1, 0)\) supersymmetric gauge theory and its minimal coupling to a hypermultiplet. In components, the kinetic term for the gauge field in such a theory involves four space-time derivatives. The theory is quantized in the framework of the superfield background method ensuring manifest \(6D\), \(\mathcal{N} = (1, 0)\) supersymmetry and the classical gauge invariance of the quantum effective action. We evaluate the superficial degree of divergence and prove it to be independent of the number of loops. Using the regularization by dimensional reduction, we find possible counterterms and show that they can be removed by the coupling constant renormalization for any number of loops, while the divergences in the hypermultiplet sector are absent at all. Assuming that the deviation of the gauge-fixing term from that in the Feynman gauge is small, we explicitly calculate the divergent part of the one-loop effective action in the lowest order in this deviation. In the approximation considered, the result is independent of the gauge-fixing parameter and agrees with the earlier calculation for the theory without a hypermultiplet.Quasinormal modes in extremal Reissner-Nordström spacetimeshttps://zbmath.org/1472.811832021-11-25T18:46:10.358925Z"Gajic, Dejan"https://zbmath.org/authors/?q=ai:gajic.dejan"Warnick, Claude"https://zbmath.org/authors/?q=ai:warnick.claude-mAuthors' abstract: We present a new framework for characterizing quasinormal modes (QNMs) or resonant states for the wave equation on asymptotically flat spacetimes, applied to the setting of extremal Reissner-Nordström black holes. We show that QNMs can be interpreted as honest eigenfunctions of generators of time translations acting on Hilbert spaces of initial data, corresponding to a suitable time slicing. The main difficulty that is present in the asymptotically flat setting, but is absent in the previously studied asymptotically de Sitter or anti de Sitter sub-extremal black hole spacetimes, is that \(L^2\)-based Sobolev spaces are not suitable Hilbert space choices. Instead, we consider Hilbert spaces of functions that are additionally Gevrey regular at infinity and at the event horizon. We introduce \(L^2\)-based Gevrey estimates for the wave equation that are intimately connected to the existence of conserved quantities along null infinity and the event horizon. We relate this new framework to the traditional interpretation of quasinormal frequencies as poles of the meromorphic continuation of a resolvent operator and obtain new quantitative results in this setting.Twisted differential \(K\)-characters and D-braneshttps://zbmath.org/1472.811922021-11-25T18:46:10.358925Z"Ferrari Ruffino, Fabio"https://zbmath.org/authors/?q=ai:ferrari-ruffino.fabio"Rocha Barriga, Juan Carlos"https://zbmath.org/authors/?q=ai:rocha-barriga.juan-carlosSummary: We analyse in detail the language of partially non-abelian Deligne cohomology and of twisted differential \(K\)-theory, in order to describe the geometry of type II superstring backgrounds with D-branes. This description will also provide the opportunity to show some mathematical results of independent interest. In particular, we begin classifying the possible gauge theories on a D-brane or on a stack of D-branes using the intrinsic tool of long exact sequences. Afterwards, we recall how to construct two relevant models of differential twisted \(K\)-theory, paying particular attention to the dependence on the twisting cocycle within its cohomology class. In this way we will be able to define twisted \(K\)-homology and twisted Cheeger-Simons \(K\)-characters in the category of simply-connected manifolds, eliminating any unnatural dependence on the cocycle. The ambiguity left for non simply-connected manifolds will naturally correspond to the ambiguity in the gauge theory, following the previous classification. This picture will allow for a complete characterization of D-brane world-volumes, the Wess-Zumino action and topological D-brane charges within the \(K\)-theoretical framework, that can be compared step by step to the old cohomological classification. This has already been done for backgrounds with vanishing \(B\)-field; here we remove this hypothesis.Time-space noncommutativity and Casimir effecthttps://zbmath.org/1472.812332021-11-25T18:46:10.358925Z"Harikumar, E."https://zbmath.org/authors/?q=ai:harikumar.e"Panja, Suman Kumar"https://zbmath.org/authors/?q=ai:panja.suman-kumar"Rajagopal, Vishnu"https://zbmath.org/authors/?q=ai:rajagopal.vishnuSummary: We show that the Casimir force and energy are modified in the \(\kappa\)-deformed space-time. This is shown by solving the Green's function corresponding to \(\kappa \)-deformed scalar field equation in presence of two parallel plates, modelled by \(\delta\)-function potentials. Exploiting the relation between Energy-Momentum tensor and Green's function, we calculate corrections to Casimir force, valid up to second order in the deformation parameter. The Casimir force is shown to get corrections which scale as \(L^{- 4}\) and \(L^{- 6}\) and both these types of corrections produce attractive forces. Using the measured value of Casimir force, we show that the deformation parameter should be below \(10^{-23}\) m.Towards an axiomatic formulation of noncommutative quantum field theory. II.https://zbmath.org/1472.812472021-11-25T18:46:10.358925Z"Chaichian, M."https://zbmath.org/authors/?q=ai:chaichian.masud"Mnatsakanova, M. N."https://zbmath.org/authors/?q=ai:mnatsakanova.m-n"Vernov, Yu. S."https://zbmath.org/authors/?q=ai:vernov.yu-sSummary: Classical results of the axiomatic quantum field theory -- irreducibility of the set of field operators, Reeh and Schlieder's theorems and generalized Haag's theorem are proven in \(SO(1, 1)\) invariant quantum field theory, of which an important example is noncommutative quantum field theory. In \(SO(1, 3)\) invariant theory new consequences of generalized Haag's theorem are obtained. It has been proven that the equality of four-point Wightman functions in two theories leads to the equality of elastic scattering amplitudes and thus the total cross-sections in these theories.
For Part I, see [the first author et al., J. Math. Phys. 52, No. 3, 032303, 13 p. (2011; Zbl 1315.81096)].Scattering, spectrum and resonance states completeness for a quantum graph with Rashba Hamiltonianhttps://zbmath.org/1472.812512021-11-25T18:46:10.358925Z"Blinova, Irina V."https://zbmath.org/authors/?q=ai:blinova.irina-v"Popov, Igor Y."https://zbmath.org/authors/?q=ai:popov.igor-yu"Smolkina, Maria O."https://zbmath.org/authors/?q=ai:smolkina.maria-oSummary: Quantum graphs consisting of a ring with two semi-infinite edges attached to the same point of the ring is considered. We deal with the Rashba spin-orbit Hamiltonian on the graph. A theorem concerning to completeness of the resonance states on the ring is proved. Due to use of a functional model, the problem reduces to factorization of the characteristic matrix-function. The result is compared with the corresponding completeness theorem for the Schrödinger, Dirac and Landau quantum graphs.
For the entire collection see [Zbl 1471.47002].Metastability of the proximal point algorithm with multi-parametershttps://zbmath.org/1472.900862021-11-25T18:46:10.358925Z"Dinis, Bruno"https://zbmath.org/authors/?q=ai:dinis.bruno"Pinto, Pedro"https://zbmath.org/authors/?q=ai:pinto.pedro-cSummary: In this article we use techniques of proof mining to analyse a result, due to \textit{Y. Yao} and \textit{M. A. Noor} [J. Comput. Appl. Math. 217, No. 1, 46--55 (2008; Zbl 1147.65049)], concerning the strong convergence of a generalized proximal point algorithm which involves multiple parameters. Yao and Noor's result [loc. cit.] ensures the strong convergence of the algorithm to the nearest projection point onto the set of zeros of the operator. Our quantitative analysis, guided by \textit{F. Ferreira} and \textit{P. Oliva}'s [Ann. Pure Appl. Logic 135, No. 1--3, 73--112 (2005; Zbl 1095.03060)] bounded functional interpretation, provides a primitive recursive bound on the metastability for the convergence of the algorithm, in the sense of Terence Tao. Furthermore, we obtain quantitative information on the asymptotic regularity of the iteration. The results of this paper are made possible by an arithmetization of the lim sup.A system-optimization model for multiclass human migration with migration costs and regulations inspired by the Covid-19 pandemichttps://zbmath.org/1472.900962021-11-25T18:46:10.358925Z"Cappello, Giorgia"https://zbmath.org/authors/?q=ai:cappello.giorgia"Daniele, Patrizia"https://zbmath.org/authors/?q=ai:daniele.patrizia"Nagurney, Anna"https://zbmath.org/authors/?q=ai:nagurney.annaSummary: In the last several decades, the main causes of human migration have included: poverty, war and political strife, climate change, tsunamis, earthquakes, as well as economic and educational possibilities. In this paper, we present a system-optimized network model for multiple migration classes with migration costs and regulations inspired by the Covid-19 pandemic. We derive the variational inequality formulation associated with the system-optimization problem which consists of maximizing the total societal welfare. Lagrange analysis is also performed in order to obtain a precise evaluation of the multiclass human migration phenomenon. This work adds to the literature on system-optimization of human migration in the presence of regulations and with the explicit inclusion of migration costs.Tensor representation of topographically organized semantic spaceshttps://zbmath.org/1472.920382021-11-25T18:46:10.358925Z"Pomi, Andrés"https://zbmath.org/authors/?q=ai:pomi.andres"Mizraji, Eduardo"https://zbmath.org/authors/?q=ai:mizraji.eduardo"Lin, Juan"https://zbmath.org/authors/?q=ai:lin.juanSummary: Human brains seem to represent categories of objects and actions as locations in a continuous semantic space across the cortical surface that reflects the similarity among categories. This vision of the semantic organization of information in the brain, suggested by recent experimental findings, is in harmony with the well-known topographically organized somatotopic, retinotopic, and tonotopic maps in the cerebral cortex. Here we show that these topographies can be operationally represented with context-dependent associative memories. In these models, the input vectors and, eventually also, the associated output vectors are multiplied by context vectors via the Kronecker tensor product, which allows a spatial organization of memories. Input and output tensor contexts localize matrices of semantic categories into a neural layer or slice and, at the same time, direct the flow of information arriving at the layer to a specific address, and then forward the output information toward the corresponding targets. Given a neural topographic pattern, the tensor representation will place a set of associative matrix memories within a topographic regionalized host matrix in such way that they reproduce the empirical pattern of patches in the actual neural layer. Progressive approximations to this goal are accomplished by avoiding excessive overlap of memories or the existence of empty regions within the host matrix.