Recent zbMATH articles in MSC 46https://zbmath.org/atom/cc/462021-01-08T12:24:00+00:00WerkzeugOn a compactness criteria for multidimensional Hardy type operator in \(p\)-convex Banach function spaces.https://zbmath.org/1449.470682021-01-08T12:24:00+00:00"Bandaliev, R. A."https://zbmath.org/authors/?q=ai:bandaliev.rovshan-alifaga-oglySummary: The main goal of this paper is to prove a criteria on compactness of a multidimensional Hardy type operator from weighted Lebesgue spaces into \(p\)-convex weighted Banach function spaces. The analogous problem for the dual operator is considered.Some equalities and inequalities for Parseval \(K\)-frames.https://zbmath.org/1449.420562021-01-08T12:24:00+00:00"Fu, Yuankang"https://zbmath.org/authors/?q=ai:fu.yuankang"Zhu, Yucan"https://zbmath.org/authors/?q=ai:zhu.yucanSummary: We give the equalities and inequalities for Parseval \(K\)-frames by using a conclusion about the equalities and inequalities for \(K\)-frames. Then we introduce the concept of upper and lower indexes of Parseval \(K\)-frames and discuss some properties of upper and lower indexes of Parseval \(K\)-frames. The conclusions given in this paper generalize the corresponding conclusions.The Jacobson radical of certain semicrossed products.https://zbmath.org/1449.471322021-01-08T12:24:00+00:00"Wiart, Jaspar"https://zbmath.org/authors/?q=ai:wiart.jasparSummary: We study the Jacobson radical of the semicrossed product \(A\times_\alpha P\) when \(\mathcal{A}\) is a simple \(C^*\)-algebra and \(P\) is either a semigroup contained in an abelian group or a free semigroup. A full characterization is obtained for a large subset of these semicrossed products and we apply our results to a number of examples.On the structure of universal differentiability sets.https://zbmath.org/1449.460342021-01-08T12:24:00+00:00"Dymond, Michael"https://zbmath.org/authors/?q=ai:dymond.michaelSummary: A subset of \(\mathbb{R}^d\) is called a universal differentiability set if it contains a point of differentiability of every Lipschitz function \(f\colon\mathbb{R}^d\to\mathbb{R}\). We show that any universal differentiability set contains a `kernel' in which the points of differentiability of each Lipschitz function are dense. We further prove that no universal differentiability set may be decomposed as a countable union of relatively closed, non-universal differentiability sets.Survey on the Kakutani problem in \(p\)-adic analysis. I.https://zbmath.org/1449.120022021-01-08T12:24:00+00:00"Escassut, Alain"https://zbmath.org/authors/?q=ai:escassut.alainSummary: Let \(\mathbb{K}\) be a complete ultrametric algebraically closed field and let \(A\) be the Banach \(\mathbb{K}\)-algebra of bounded analytic functions in the ``open'' unit disk \(D\) of \(\mathbb{K}\) provided with the Gauss norm. Let \(\operatorname{Mult}(A,\|.\|)\) be the set of continuous multiplicative semi-norms of \(A\) provided with the topology of pointwise convergence, let \(\operatorname{Mult}_m(A,\|.\|)\) be the subset of the \(\Phi \in \operatorname{Mult}(A,\|.\|)\) whose kernel is a maximal ideal and let \(\operatorname{Mult}_1(A,\|.\|)\) be the subset of the \(\Phi \in \operatorname{Mult}(A,\|.\|)\) whose kernel is a maximal ideal of the form \((x - a)A\) with \(a \in D\). By analogy with the Archimedean context, one usually calls ultrametric corona problem, or ultrametric Kakutani problem the question whether \(\operatorname{Mult}_1(A,\|.\|)\) is dense in \(\operatorname{Mult}_m(A,\|.\|)\). In order to recall the study of this problem that was made in several successive steps, here we first recall how to characterize the various continuous multiplicative semi-norms of \(A\), with particularly the nice construction of certain multiplicative semi-norms of \(A\) whose kernel is neither a null ideal nor a maximal ideal, due to J. Araujo. Here we prove that multbijectivity implies density. The problem of multbijectivity will be described in a further paper [Part II, Sarajevo J. Math. 16(29), No. 1, 55--70 (2020; Zbl 07258265)].Quasi-asymptotically almost periodic vector-valued generalized functions.https://zbmath.org/1449.460302021-01-08T12:24:00+00:00"Kostić, Marko"https://zbmath.org/authors/?q=ai:kostic.marko"Pilipović, Stevan"https://zbmath.org/authors/?q=ai:pilipovic.stevan-r"Velinov, Daniel"https://zbmath.org/authors/?q=ai:velinov.danielSummary: In this paper are introduced the notions of quasi-asymptotically almost periodic distributions and quasi-asymptotically almost periodic ultradistributions with values in a Banach space, as well as some other generalizations of these concepts. Furthermore, some applications of the introduced concepts in the analysis of systems of ordinary differential equations are provided.Essential spectrum and Fredholm properties for operators on locally compact groups.https://zbmath.org/1449.460582021-01-08T12:24:00+00:00"Măntoiu, Marius Laurenţiu"https://zbmath.org/authors/?q=ai:mantoiu.mariusSummary: We study the essential spectrum and Fredholm properties of certain integral and pseudo-differential operators associated to non-commutative locally compact groups~$G$. The techniques involve crossed product \(C^*\)-algebras. We extend previous results on the structure of the essential spectrum to self-adjoint operators belonging (or affiliated) to the Schrödinger representation of certain crossed products. When the group $G$ is unimodular and type~I, we cover a new class of global pseudo-differential differential operators with operator-valued symbols involving the unitary dual of~$G$. We use recent results of Nistor, Prudhon and Roch on the role of families of representations in spectral theory and the notion of quasi-regular dynamical system.Almost uniform and strong convergences in ergodic theorems for symmetric spaces.https://zbmath.org/1449.470242021-01-08T12:24:00+00:00"Chilin, V."https://zbmath.org/authors/?q=ai:chilin.vladimir-ivanovich|chilin.vladmir"Litvinov, S."https://zbmath.org/authors/?q=ai:litvinov.s-v|litvinov.semyon-n|litvinov.sergej|litvinov.s-a|litvinov.sergeyA space $X\subset L^0_\nu$ is fully symmetric on $((0,\infty),\nu)$ if $f\in X$, $g\in L^0_\nu$, and the decreasing rearrangement of $f$ dominates that of $g: g^\ast\leq f^\ast$ pointwise (resp., $\int_0^s g^\ast (t)\, dt\leq \int_0^s f^\ast(t)\, dt$ for all $s>0$) implies that $g\in X$ and $\Vert g\Vert_X\leq \Vert f\Vert_X$.
The first main result extends the Dunford-Schwartz pointwise ergodic theorem in characterizing $\mathcal{R}_\mu=\{f\in L^1+L^\infty: \forall\lambda>0,\, \mu(\vert f\vert >\lambda)<\infty\}$ as a space on which pointwise ergodic limits converge uniformly. To be precise, let $(\Omega,\mathcal{A},\mu)$ be a measure space and $X$ a fully symmetric space on $(\Omega,\mathcal{A},\mu)$ such that the constant $1\notin X$. If $T\in DS$ (that is, $T$ is bounded on both $L^1$ and $L^\infty$) and $f\in X$, then the averages $M_n(T)(f)=\frac{1}{n}\sum_{k=0}^{n-1} T^k(f)$ converge a.u. to some $\hat{f}\in X$. In particular, $M_n(T)(f) =\frac{1}{n}\sum_{k=0}^{n-1} T^k (f)\, \to\hat{f}\in \mathcal{R}_\mu$ a.u. when $f\in \mathcal{R}_\mu$.
The proofs involve a reduction to the $L^1$ case and the Hopf maximal theorem ($\int_{M(T)^\ast(f)>0} f\, d\mu>0$), where $M(T)^\ast(f)(x)=\sup \vert M_n(T)(f)(x)\vert $, and the weak type inequalities $$\mu(M(T)^\ast(\vert f\vert)>\lambda)\leq \bigl(2\frac{\Vert f\Vert_p}{\lambda}\bigr)^p,\quad\lambda>0\, .$$
Theorem 3.4 then states that, if $\mu$ is $\sigma$-finite, then $\mathcal{R}_\mu$ is the largest subspace of $L^1+L^\infty$ on which convergence is almost uniform, that is, if $f\in (L^1+L^\infty)\setminus \mathcal{R}_\mu$, then there is a $T\in DS$ such that the sequence $M_n(T)(f)$ fails to converge almost everywhere. In fact, the maximality of $\mathcal{R}_\mu$ for a subspace $X$ is equivalent to constants not belonging to $X$. Orlicz spaces are used to illustrate the condition that constants are not members. Strong convergence of Cesàro means is also discussed in the context of characterizing validity of mean ergodicity for fully symmetric spaces.
Reviewer: Joseph Lakey (Las Cruces)Constructions of \(K\)-frames in Hilbert spaces.https://zbmath.org/1449.420542021-01-08T12:24:00+00:00"Du, Dandan"https://zbmath.org/authors/?q=ai:du.dandan"Zhu, Yucan"https://zbmath.org/authors/?q=ai:zhu.yucanSummary: A \(K\)-frame is a generalization of a frame in a Hilbert space. In this paper we use two Bessel sequences to construct a \(K\)-frame, a \({T_1}\)-frame or \({T_2}\)-frame in a Hilbert space. We also construct a \(P\)-frame or \(Q\)-frame by two \(K\)-frames in a Hilbert space. Our results generalize and improve the existing remarkable results.\(Tb\) criteria for Calderón-Zygmund operators on Lipschitz spaces with para-accretive functions.https://zbmath.org/1449.420282021-01-08T12:24:00+00:00"Zheng, Taotao"https://zbmath.org/authors/?q=ai:zheng.taotao"Tao, Xiangxing"https://zbmath.org/authors/?q=ai:tao.xiangxingThe goal of this paper is to give a Tb criteria for the boundedness of Calderón-Zygmund operators on the Lipschitz spaces \[\mathrm{Lip}_b(\alpha)(\mathbb R^n).\] The main machine is to develop the Littlewood-Paley characterization for Lipschitz spaces \(\mathrm{Lip}_b(\alpha)(\mathbb R^n)\) and \(\mathrm{Lip}(\alpha)(\mathbb R^n)\), which also has its own value and significance.
Then, the authors prove that the Calderón-Zygmund operators \(T\) are bounded from \(\mathrm{Lip}_b(\alpha)(\mathbb R^n)\) to \(\mathrm{Lip}(\alpha)(\mathbb R^n)\) if and only if \[T(b) \,= \,0.\] For this purpose, the authors recall the useful tools that are the para-accretive function, test function spaces, an approximation to the identity and some discrete versions of the Calderón type reproducing formula.
Finally, it is pointed out that to prove the main result in the paper, the authors need the boundedness of Calderón-Zygmund operator on Hardy spaces.
Reviewer: Maria Alessandra Ragusa (Catania)Non-global nonlinear Lie triple derivable maps on factor von Nuemann algebras.https://zbmath.org/1449.160852021-01-08T12:24:00+00:00"Su, Yutian"https://zbmath.org/authors/?q=ai:su.yutian"Zhang, Jianhua"https://zbmath.org/authors/?q=ai:zhang.jianhua|zhang.jianhua.1Summary: Let \(M\) be a factor von Neumann algebra with dimension greater than 1 on a Hilbert space \(H\). With the help of algebraic decomposition method, we prove that if a nonlinear map \(\delta:M \to M\) satisfied \(\delta ([[A, B], C]) = [[\delta (A), B], C] + [[A, \delta (B)], C] + [[A, B], \delta (C)]\) for any \(A\), \(B\), \(C \in M\) with \(ABC = 0\), then there existed an additive derivation \(d:M \to M\), such that \(\delta (A) = d (A) + \tau (A)I\) for any \(A \in M\), where \(\tau :M \to \mathbb{C}I\) is a nonlinear map such that \(\tau ([[A, B], C]) = 0\) with \(ABC = 0\) for any \(A\), \(B\), \(C \in M\).Quasimonotonicity and functional inequalities.https://zbmath.org/1449.354642021-01-08T12:24:00+00:00"Herzog, Gerd"https://zbmath.org/authors/?q=ai:herzog.gerd"Volkmann, Peter"https://zbmath.org/authors/?q=ai:volkmann.peterSummary: A comparison theorem for functional equations in ordered topological vector spaces will be given, which generalizes the results from \textit{P. Volkmann} [ISNM, Int. Ser. Numer. Math. 161, 269--273 (2012; Zbl 1253.26043); Ein Vergleichssatz für Integralgleichungen, KITopen, 3 p. (2016; \url{doi:10.5445/IR/1000061837})]. Quasimonotonicity is fundamental for these investigations.Caratheodory-Hahn-Kluvanek extension theorem on locally convex space.https://zbmath.org/1449.460012021-01-08T12:24:00+00:00"Wurenqiqige"https://zbmath.org/authors/?q=ai:wurenqiqige."Yang, Meirong"https://zbmath.org/authors/?q=ai:yang.meirongSummary: Using the method of vector measure theory in Banach space, we further study the Caratheodory-Hahn-Kluvanek extension theorem on the locally convex space. First we introduce the concepts of \(P\) completeness and separation in locally convex space. We discuss the Caratheodory-Hahn-Kluvanek extension theorem on \(P\)-completed locally convex separated space, and further give the Caratheodory-Hahn-Kluvanek extension theorem on the product space of Banach spaces.The duality of \(K\)-frames in Hilbert \(C^*\)-modules.https://zbmath.org/1449.460502021-01-08T12:24:00+00:00"Xiang, Zhongqi"https://zbmath.org/authors/?q=ai:xiang.zhongqi"Shi, Huangping"https://zbmath.org/authors/?q=ai:shi.huangpingSummary: The present paper studies the dual problems of \(K\)-frames in Hilbert \(C^*\)-modules. Some characterizations for \(K\)-dual Bessel sequences in Hilbert \(C^*\)-modules are obtained by using the operator theory methods, which generalize the duality theory of \(K\)-frames in Hilbert spaces.Some results on \(L\)-weakly compact sets and operators.https://zbmath.org/1449.470782021-01-08T12:24:00+00:00"Lhaimer, Driss"https://zbmath.org/authors/?q=ai:lhaimer.driss"Bouras, Khalid"https://zbmath.org/authors/?q=ai:bouras.khalid"Moussa, Mohammed"https://zbmath.org/authors/?q=ai:moussa.mohammedSummary: In this paper, we give some new characterizations of \(L\)-weakly compact operators. As consequences, we will give some interesting results. Also, we establish some necessary and sufficient conditions on which a relatively weakly compact set is \(L\)-weakly compact. In particular, we characterize Banach lattices with the positive Schur property.Sums of \(K\)-Riesz frames in Hilbert spaces.https://zbmath.org/1449.420572021-01-08T12:24:00+00:00"Huang, Xinli"https://zbmath.org/authors/?q=ai:huang.xinli"Zhu, Yucan"https://zbmath.org/authors/?q=ai:zhu.yucanSummary: According to the operator theory, the paper suggests a new research method to solve the problem of how the sum of a \(K\)-Riesz frame and a sequence generates a new \(K\)-Riesz frame. Then the sufficient conditions for generating a new \(K\)-Riesz frame are obtained. The results in the paper correct the remarkable results of Riesz-frames.Approximately 2-local derivations on the semi-finite von Neumann algebras.https://zbmath.org/1449.470722021-01-08T12:24:00+00:00"Zhao, Xingpeng"https://zbmath.org/authors/?q=ai:zhao.xingpeng"Fang, Xiaochun"https://zbmath.org/authors/?q=ai:fang.xiaochun"Yang, Bing"https://zbmath.org/authors/?q=ai:yang.bingSummary: The definition of approximately 2-local derivation on von Neumann algebras is introduced based on the definitions of approximately local derivation and 2-local derivation. Approximately 2-local derivations on semi-finite von Neumann algebras are studied. Let \(M\) be a von Neumann algebra and \(\Delta: {M} \to {M}\) be an approximately 2-local derivation. It is easy to obtain that \(\Delta\) is homogeneous and \(\Delta\) satisfies \(\Delta ({x^2}) = \Delta (x)x + x\Delta (x)\) for any \(x \in {M}\). Besides, if \(M\) is a von Neumann algebra with a faithful normal semi-finite trace \(\tau\), then a sufficient condition for \(\Delta\) to be additive is given, that is, \(\Delta ({M}_\tau) \subseteq {M}_\tau\), where \(M_\tau = \{x \in M:\tau (|x|) < \infty\}\). In all, if \(\Delta\) is an approximately 2-local derivation on a semi-finite von Neumann algebra with a faithful normal semi-finite trace \(\tau\) and satisfies \(\Delta (M_\tau) \subseteq M_\tau\), where \(M_\tau = \{x \in M\}:\tau (|x|) < \infty\}\), by the conclusion that the Jordon derivation from a 2-torsion free semi-prime ring to itself is a derivation, it follows that \(\Delta\) is a derivation.Boundary representations of operator spaces, and compact rectangular matrix convex sets.https://zbmath.org/1449.460472021-01-08T12:24:00+00:00"Fuller, Adam H."https://zbmath.org/authors/?q=ai:fuller.adam-hanley"Hartz, Michael"https://zbmath.org/authors/?q=ai:hartz.michael"Lupini, Martino"https://zbmath.org/authors/?q=ai:lupini.martinoSummary: We initiate the study of matrix convexity for operator spaces. We define the notion of compact rectangular matrix convex set, and prove the natural analogs of the Krein-Milman and the bipolar theorems in this context. We deduce a canonical correspondence between compact rectangular matrix convex sets and operator spaces. We also introduce the notion of boundary representation for an operator space, and prove the natural analog of Arveson's conjecture: every operator space is completely normed by its boundary representations. This yields a canonical construction of the triple envelope of an operator space.\(f\)-orthomorphisms and \(f\)-linear operators on the order dual of an \(f\)-algebra revisited.https://zbmath.org/1449.460032021-01-08T12:24:00+00:00"Jaber, Jamel"https://zbmath.org/authors/?q=ai:jaber.jamelSummary: We give a necessary and sufficient condition on an \(f\)-algebra \(A\) for which orthomorphisms, \(f\)-linear operators, and \(f\)-orthomorphisms on the order dual \(A^\sim\) are the same class of operators.Commutators of bilinear \(\theta\)-type Calderón-Zygmund operators on Morrey spaces over non-homogeneous spaces.https://zbmath.org/1449.420162021-01-08T12:24:00+00:00"Lu, G.-H."https://zbmath.org/authors/?q=ai:lu.gui-hua|lu.guohao|lu.genghong|lu.guanhua|lu.guanghui|lu.guang-hongThe author proves some boundedness properties for commutators which are generated by the bilinear \(\theta\)-type Calderón-Zygmund operators \(T_\theta\) and two functions \(b_1\), \(b_2\) belonging to the space that is a variant of the bounded mean oscillation class, firstly defined by \textit{F. John} and \textit{L. Nirenberg} [Commun. Pure Appl. Math. 14, 415--426 (1961; Zbl 0102.04302)].
Precisely, \([b_1,b_2,T_\theta]\) is bounded from the Lebesgue space \(L^p(\mu)\) into the product of Lebesgue spaces \[L^{p_1}(\mu) \times L^{p_2}(\mu), \quad \frac{1}{p}\,=\, \frac{1}{p_1}\,+\, \frac{1}{p_2}, \,\,(1 < p, p_1, p_2 < \infty)\] being \(\mu\) a Borel measure.
Moreover the boundedness of the commutator \([b_1, b_2, T_\theta]\) on the Morrey space \(M^q_p(\mu)\), \(1\,<\,q\,<\,p\,<\,\infty\) is obtained. Main tools are the definitions of geometrically doubling metric space and upper doubling metric measure space.
Reviewer: Maria Alessandra Ragusa (Catania)Harmonic polynomials via differentiation.https://zbmath.org/1449.460312021-01-08T12:24:00+00:00"Estrada, Ricardo"https://zbmath.org/authors/?q=ai:estrada.ricardoSummary: It is well-known that if \(p\) is a homogeneous polynomial of degree \(k\) in \(n\) variables, \(p \in {\mathcal{P}_k}\), then the ordinary derivative \(p\left(\nabla \right)\left({{r^{2 - n}}} \right)\) has the form \({A_{n, k}}\mathcal{Y}\left(x \right){r^{2 - n - 2k}}\) where \({A_{n, k}}\) is a constant and \(\mathcal{Y}\) is a harmonic homogeneous polynomial of degree \(k\), \(\mathcal{Y} \in {\mathcal{H}_k}\), actually the projection of \(p\) onto \({\mathcal{H}_k}\). Here we study the distributional derivative \(p\left({\bar \nabla} \right)\left({{r^{2 - n}}} \right)\) and show that the ordinary part is still a multiple of \(\mathcal{Y}\), but that the delta part is independent of \(\mathcal{Y}\), that is, it depends only on \(p - \mathcal{Y}\). We also show that the exponent \(2 - n\) is special in the sense that the corresponding results for \(p\left(\nabla \right)\left({{r^\alpha}} \right)\) do not hold if \(\alpha \ne 2 - n\). Furthermore, we establish that harmonic polynomials appear as multiples of \({r^{2 - n - 2k - 2k'}}\) when \(p\left(\nabla \right)\) is applied to harmonic multipoles of the form \(\mathcal{Y}'\left(x \right){r^{2 - n - 2k'}}\) for some \(\mathcal{Y}' \in {\mathcal{H}_k}\).Realization of rigid \(C^*\)-tensor categories via Tomita bimodules.https://zbmath.org/1449.460482021-01-08T12:24:00+00:00"Giorgetti, Luca"https://zbmath.org/authors/?q=ai:giorgetti.luca"Yuan, Wei"https://zbmath.org/authors/?q=ai:yuan.weiLet \({\mathfrak{A}}\) be a von Neumann algebra, and \(H\) denotes a pre-Hilbert \({\mathfrak{A}}$-${\mathfrak{A}}\) bimodule. \(\Omega \in H\) is a distinguished vacuum vector such that \(\langle \Omega \vert \Omega \rangle_{ {\mathfrak{A}}} = I\) and \(A \cdot \Omega\) \(=\) \(\Omega \cdot A\) for every \(A \in {\mathfrak{A}}\). When \({\mathfrak{A}}\) is a semifinite von Neumann algebra and \(\tau\) is a normal semifinite faithful tracial weight on \({\mathfrak{A}}\), then \[ {\mathfrak{N}}({\mathfrak{A}}, \tau) = \{ A \in {\mathfrak{A}}; \, \tau( A^* A) < \infty \, \} \] is an ideal. \(H\) is said to be a Tomita \({\mathfrak{A}}$-${\mathfrak{A}}\) bimodule if (i) \({\mathfrak{N}}( H, \tau)\) \(=\) \(H\); (ii) \(H\) admits an involution \(S\) such that \(S( A \cdot \zeta \cdot B)\) \(=\) \(B^* \cdot S(\zeta) \cdot A^*\) for every \(\zeta\in H\) and \(A, B \in {\mathfrak{A}}\); (iii) \(H\) admits a complex one-parameter group \(\{ {\mathcal{U}}(\alpha); \, \alpha \in {\mathbb{C}} \, \}\) of linear isomorphisms satisfying some intertwining properties between \(S\) and \({\mathcal{U}}\). \({\mathcal{B}}(H)\) is the set of bounded adjointable linear mappings from \(H\) to \(H\). \({\mathcal{F}}(H)\) is a Fock space in Voiculescu's free probability theory, associated to a Tomita \({\mathfrak{A}}$-${\mathfrak{A}}\) bimodule \(H\). Note that \({\mathcal{F}}(H)\) is a pre-Hilbert \({\mathfrak{A}}$-${\mathfrak{A}}\) bimodule. Let \(\Phi(H)\) be the \(*\)-subalgebra of \({\mathcal{B}}( {\mathcal{F}}(H))\) generated by \({\mathfrak{A}}\) acting on \({\mathcal{F}}(H)\) from the left, and let \(\Phi(H)''\) be the von Neumann algebra generated by \(\pi_{\tau}( \Phi(H))\) on the Hilbert space completion \({\mathcal{F}}(H)_{\tau}\), and let \(\Phi(H)'\) be the commutant of \(\Phi(H)''\). Note that \(\Phi(H) \Omega ={\mathcal{F}}(H)\).
This paper treats a construction of von Neumann algebras \(\Phi(H)''\), starting from a rigid \(C^*\)-tensor category \({\mathcal{C}}\) with simple unit. Actually, these algebras \(\Phi(H)''\) are factors of type II or type III$_{\lambda}\) with \(\lambda \in (0, 1]\). As a matter of fact, the choice of type is tuned by the choice of Tomita structure on certain bimodules of which the authors make use in the construction. In the special case where the spectrum is infinite, the whole tensor category can be realized as endomorphisms of these algebras. Furthermore, it is clarified that, if the Tomita structure is trivial, the algebras obtained in this paper are amplifications of the free group factors with infinitely many generators.
Reviewer: Isamu Dôku (Saitama)On the geometry of the unit ball of a \(J B^{*}\)-triple.https://zbmath.org/1449.460612021-01-08T12:24:00+00:00"Tahlawi, Haifa M."https://zbmath.org/authors/?q=ai:tahlawi.haifa-m"Siddiqui, Akhlaq A."https://zbmath.org/authors/?q=ai:siddiqui.akhlaq-ahmad"Jamjoom, Fatmah B."https://zbmath.org/authors/?q=ai:jamjoom.fatmah-backerSummary: We explore a \(J B^{*}\)-triple analogue of the notion of quasi invertible elements, originally studied by Brown and Pedersen in the setting of \(C^*\)-algebras. This class of BP-quasi invertible elements properly includes all invertible elements and all extreme points of the unit ball and is properly included in von Neumann regular elements in a \(J B^*\)-triple; this indicates their structural richness. We initiate a study of the unit ball of a \(J B^*\)-triple investigating some structural properties of the BP-quasi invertible elements; here and in sequent papers, we show that various results on unitary convex decompositions and regular approximations can be extended to the setting of BP-quasi invertible elements. Some \(C^*\)-algebra and \(J B^*\)-algebra results, due to Kadison and Pedersen, Rørdam, Brown, Wright and Youngson, and Siddiqui, including the Russo-Dye theorem, are extended to \(J B^*\)-triples.\(p\)-convergent operators and the \(p\)-Schur property.https://zbmath.org/1449.460212021-01-08T12:24:00+00:00"Alikhani, M."https://zbmath.org/authors/?q=ai:alikhani.malihe|alikhani.mahdi|alikhani.morteza|alikhani.masoomeh"Fakhar, M."https://zbmath.org/authors/?q=ai:fakhar.majid"Zafarani, J."https://zbmath.org/authors/?q=ai:zafarani.jafarLet \(X,Y\) be Banach spaces. This interesting paper deals with the question of certain subclasses of \({\mathcal{K}}(X,Y)\) being equal to \({\mathcal{K}}(X,Y)\) for all spaces \(Y\) or equivalently for a test space \(Y\).
For \(1 \leq p < \infty\) let \(C_p(X,Y)\) denote the class of operators which map weakly \(p\)-summable sequences to norm null sequences. It is shown that when \(C_p(X,Y) = {\mathcal{K}}(X,Y)\) for \(Y = \ell^{\infty}\), then these spaces are equal for all Banach spaces \(Y\).
For \(1 \leq p <q < \infty\), a similar result holds for the inclusion \(C_p(X,Y) \subset C_q(X,Y)\).
Reviewer: T.S.S.R.K. Rao (Bangalore)Arithmetic summable sequence space over non-Newtonian field.https://zbmath.org/1449.260022021-01-08T12:24:00+00:00"Yaying, Taja"https://zbmath.org/authors/?q=ai:yaying.taja"Hazarika, Bipan"https://zbmath.org/authors/?q=ai:hazarika.bipanSummary: In this article, we introduce the sequence spaces \(AS (G)\) and \(AC (G)\) of arithmetic summable and arithmetic convergent sequences, respectively, suggested by the geometric sum \(_G\sum_{k|m} f(k)\) as \(k\) ranges over the divisors of \(m\). We further obtain an analogous of Möbius inversion formula in the sense of geometric calculus and give interesting results in the geometric field.Ball-covering property in uniformly non-\(l_3^{(1)}\) Banach spaces and application.https://zbmath.org/1449.460182021-01-08T12:24:00+00:00"Shang, Shaoqiang"https://zbmath.org/authors/?q=ai:shang.shaoqiang"Cui, Yunan"https://zbmath.org/authors/?q=ai:cui.yunanSummary: This paper shows the following. (1) \(X\) is a uniformly non-\(l_3^{(1)}\) space if and only if there exist two constants \(\alpha, \beta > 0\) such that, for every 3-dimensional subspace \(Y\) of \(X\), there exists a ball-covering \(\mathfrak{B}\) of \(Y\) with \(c(\mathfrak{B}) = 4\) or \(5\) which is \(\alpha\)-off the origin and \(r(\mathfrak{B}) \leq \beta\). (2) If a separable space \(X\) has the Radon-Nikodým property, then \(X^*\) has the ball-covering property. Using this general result, we find sufficient conditions in order that an Orlicz function space has the ball-covering property.Metaplectic transformations and finite group actions on noncommutative tori.https://zbmath.org/1449.460562021-01-08T12:24:00+00:00"Chakraborty, Sayan"https://zbmath.org/authors/?q=ai:chakraborty.sayan"Luef, Franz"https://zbmath.org/authors/?q=ai:luef.franzLet \(A_\Theta\) be the \(2n\)-dimensional noncommutative torus determined by a \(2n\)-dimensional real skew-symmetric matrix \(\Theta\), and suppose \(W\in\operatorname{SL}_{2n}(\mathbb{Z})\) has order \(k\) and satisfies \(W^T\Theta W=\Theta\). The authors define the ``metaplectic'' action of \(W\) on the set \(\mathcal{S}(\mathbb{R}^n)\) of Schwartz functions, and show that this action of \(\mathbb{Z}_k\) gives rise to a finitely generated projective module over the crossed product \(C^*\)-algebra \(A_\Theta\rtimes\mathbb{Z}_k\).
Reviewer: Vladimir M. Manuilov (Moskva)Infinitesimal aspects of idempotents in Banach algebras.https://zbmath.org/1449.460362021-01-08T12:24:00+00:00"Beltiţă, Daniel"https://zbmath.org/authors/?q=ai:beltita.daniel"Galé, José E."https://zbmath.org/authors/?q=ai:gale.jose-eThe authors introduce and study Stiefel bundles on flag manifolds, which are extensions of the well known Stiefel bundles on Grassmannians. The main ingredient of the investigation is the notion of connection on an infinite-dimensional bundle. This allows to investigate infinitesimal properties of sets of ordered \(n\)-tuples of idempotents in a symmetric Banach \(*\)-algebra.
For the entire collection see [Zbl 1404.42002].
Reviewer: Cătălin Badea (Villeneuve d'Ascq)A Beurling theorem for noncommutative Hardy spaces associated with semifinite von Neumann algebras with unitarily invariant norms.https://zbmath.org/1449.460552021-01-08T12:24:00+00:00"Liu, Wenjing"https://zbmath.org/authors/?q=ai:liu.wenjing"Sager, Lauren"https://zbmath.org/authors/?q=ai:sager.lauren-b-mThe authors deal with completions \(L^\alpha({\mathcal{M}},\tau)\) under \(\alpha\) of a set of elementary operators \({\mathcal{I}}\) in a semifinite von Neumann algebra \({\mathcal{M}}\) with a faithful normal semifinite trace \(\tau\), where \(\alpha\) is a unitarily invariant norm on \({\mathcal{I}}\) satisfying some extra regularity conditions. Many examples of such norms are given. The definition of Arveson's noncommutative Hardy spaces \(H^\infty\) is extended to subspaces \(H^\alpha\) of \(L^\alpha({\mathcal{M}},\tau)\) being the closures of \({\mathcal{A}}\cap L^\alpha({\mathcal{M}}, \tau)\) with respect to the \(\alpha\)-norm, where \({\mathcal{A}}\) is a subdiagonal subalgebra of \({\mathcal{M}}\). The authors prove a Beurling-type theorem for the \(H^\alpha\) spaces, and use it to investigate invariant subspaces of a class of noncommutative Banach function spaces.
Reviewer: Stanisław Goldstein (Łódź)Cyclicity in the harmonic Dirichlet space.https://zbmath.org/1449.460242021-01-08T12:24:00+00:00"Abakumov, Evgueni"https://zbmath.org/authors/?q=ai:abakumov.evgeny-v"El-Fallah, Omar"https://zbmath.org/authors/?q=ai:el-fallah.omar"Kellay, Karim"https://zbmath.org/authors/?q=ai:kellay.karim"Ransford, Thomas"https://zbmath.org/authors/?q=ai:ransford.thomas-jSummary: The harmonic Dirichlet space is the Hilbert space of functions \(f\in L^2(\mathbb{T})\) such that \[ \Vert f\Vert^2_{\mathcal{D}(\mathbb{T})}:= \sum_{n\in\mathbb{Z}}(1+\vert n\vert)\vert\widehat f(n)\vert^2 <\infty. \] We give sufficient conditions for \(f\) to be cyclic in \(\mathcal{D}(\mathbb{T})\), that is, for \(\{\zeta^nf(\zeta):n\ge 0\}\) to span a dense subspace of \(\mathcal{D}(\mathbb{T})\).
For the entire collection see [Zbl 1404.42002].Numerical computations of nonlocal Schrödinger equations on the real line.https://zbmath.org/1449.820042021-01-08T12:24:00+00:00"Yan, Yonggui"https://zbmath.org/authors/?q=ai:yan.yonggui"Zhang, Jiwei"https://zbmath.org/authors/?q=ai:zhang.jiwei"Zheng, Chunxiong"https://zbmath.org/authors/?q=ai:zheng.chunxiongSummary: The numerical computation of nonlocal Schrödinger equations (SEs) on the whole real axis is considered. Based on the artificial boundary method, we first derive the exact artificial nonreflecting boundary conditions. For the numerical implementation, we employ the quadrature scheme proposed in [\textit{X. Tian} and \textit{Q. Du}, SIAM J. Numer. Anal. 51, No. 6, 3458--3482 (2013; Zbl 1295.82021)] to discretize the nonlocal operator, and apply the \(z\)-transform to the discrete nonlocal system in an exterior domain, and derive an exact solution expression for the discrete system. This solution expression is referred to our exact nonreflecting boundary condition and leads us to reformulate the original infinite discrete system into an equivalent finite discrete system. Meanwhile, the trapezoidal quadrature rule is introduced to discretize the contour integral involved in exact boundary conditions. Numerical examples are finally provided to demonstrate the effectiveness of our approach.Some results on second transpose of a dual valued derivation.https://zbmath.org/1449.460392021-01-08T12:24:00+00:00"Essmaili, Morteza"https://zbmath.org/authors/?q=ai:essmaili.mortezaSummary: Let \(A\) be a Banach algebra and \(X\) be an arbitrary Banach \(A\)-module. In this paper, we study the second transpose of derivations with value in dual Banach \(A\)-module \(X^*\). Indeed, for a continuous derivation \(D: A \longrightarrow X^*\) we obtain a necessary and sufficient condition such that the bounded linear map \(\Lambda \circ D^{\prime\prime} : A^{**} \longrightarrow X^{***}\) to be a derivation, where \(\Lambda\) is composition of restriction and canonical injection maps. This characterization generalizes some well known results in [\textit{M. Amini} et al., New York J. Math. 22, 265--275 (2016; Zbl 1354.46049)].Quasilinear elliptic systems in perturbed form.https://zbmath.org/1449.352262021-01-08T12:24:00+00:00"Azroul, Elhoussine"https://zbmath.org/authors/?q=ai:azroul.elhoussine"Balaadich, Farah"https://zbmath.org/authors/?q=ai:balaadich.farahSummary: In this paper, we consider the boundary value problem of a quasilinear elliptic system in degenerate form with data belongs to the dual of Sobolev spaces. The existence result is proved by means of Young measures and mild monotonicity assumptions.Sequences of contractions on cone metric spaces over Banach algebras and applications to nonlinear systems of equations and systems of differential equations.https://zbmath.org/1449.540462021-01-08T12:24:00+00:00"Alecsa, Cristian Daniel"https://zbmath.org/authors/?q=ai:alecsa.cristian-danielSummary: It is well known that fixed point problems of contractive-type mappings defined on cone metric spaces over Banach algebras are not equivalent to those in usual metric spaces (see [\textit{H. Huang} et al., Positivity 23, No. 1, 21--34 (2019; Zbl 07053419)] and [\textit{H. Liu} and \textit{S. Xu}, Fixed Point Theory Appl. 2013, Paper No. 320, 10 p. (2013; Zbl 1295.54062)]). In this framework, the novelty of the present paper represents the development of some fixed point results regarding sequences of contractions in the setting of cone metric spaces over Banach algebras. Furthermore, some examples are given in order to strengthen our new concepts. Also, based on the powerful notion of a cone metric space over a Banach algebra, we present important applications to systems of differential equations and coupled functional equations, respectively, that are linked to the concept of sequences of contractions.Topological Levinson's theorem for inverse square potentials: complex, infinite, but not exceptional.https://zbmath.org/1449.810192021-01-08T12:24:00+00:00"Inoue, H."https://zbmath.org/authors/?q=ai:inoue.hideki|inoue.hirotsugu|inoue.hitoshi|inoue.hikaru|inoue.hirochiko|inoue.hirofumi|inoue.hiroyasu|inoue.haruki|inoue.hiroe|inoue.hirotaka|inoue.hirohito|inoue.hirochika|inoue.hiroto|inoue.hironori|inoue.hideyuki|inoue.hisao|inoue.hiromi|inoue.hiroaki|inoue.hidehiko|inoue.hiroshi|inoue.hiroyuki"Richard, S."https://zbmath.org/authors/?q=ai:richard.serge|richard.sebastien|richard.scott-fThe authors consider Schrödinger operators with inverse square potentials on the half-line, and depending on some parameters. Such operators can have either a finite or an infinite number of complex eigenvalues. The Schrödinger operator of the form \[ -\partial_r^2 + \left(m^2-\frac{1}{4}\right)\frac{1}{r^2} \] defined on the half-line \(\mathbb{R}_+\) is considered. The parameter \(m\in\mathbb{C}\) with \(\mathfrak{R}(m) > -1\) is used for describing the coupling constant for the potential. For \(m\neq 0\) an additional parameter \(\kappa\in\mathbb{C}\) is used for defining the boundary condition at \(r = 0\), while for \(m = 0\) another family of operators indexed by a boundary parameter \(\nu\) is defined. The study of the corresponding families of closed operators \(H_{m,\kappa}\) and \(H_0^\nu\) in \(L^2(\mathbb{R}_+)\) has been initiated and developed by \textit{J. Dereziński} and \textit{S. Richard} [Ann. Henri Poincaré 18, No.~3, 869--928 (2017; Zbl 1370.81070)].
The main statement discussed here is Levinson's theorem. It gives a certain relation between the number of bound states of a quantum mechanical system and an expression related to the scattering part of that system. Spectral singularities embedded in the continuous spectrum which the authors call exceptional situations are considered. The spectral and the scattering theory for the above stated operators is discussed. Some new results for the exceptional cases are provided. The known index theorems in scattering theory are also considered. The question why some results cannot be extended to the exceptional cases is discussed as well.
Reviewer: Dimitar A. Kolev (Sofia)Intuitionistic fuzzy \(I\)-convergent sequence spaces defined by compact operator.https://zbmath.org/1449.460062021-01-08T12:24:00+00:00"Khan, Vakeel A."https://zbmath.org/authors/?q=ai:khan.vakeel-a"Yasmeen"https://zbmath.org/authors/?q=ai:yasmeen.k-zeba|yasmeen.saba|yasmeen.shagufta|yasmeen.farah|yasmeen.uzma"Fatima, Hira"https://zbmath.org/authors/?q=ai:fatima.hira"Altaf, Henna"https://zbmath.org/authors/?q=ai:altaf.henna"Lohani, Q. M. Danish"https://zbmath.org/authors/?q=ai:lohani.q-m-danishSummary: The purpose of this paper is to introduce the intuitionistic fuzzy \(I\)-convergent sequence spaces \(S^I_{(\mu,\nu)}(T)\) and \(S^I_{0(\mu,\nu)}(T)\) defined by compact operator and study the fuzzy topology on the above said spaces.Completely continuous Banach algebras.https://zbmath.org/1449.460382021-01-08T12:24:00+00:00"Hayati, Bahman"https://zbmath.org/authors/?q=ai:hayati.bahmanSummary: For a Banach algebra \(\mathfrak A\), we introduce \(c.c(\mathfrak A)\), the set of all \(\phi\in \mathfrak A^*\) such that \(\theta_{\phi}:\mathfrak A \to\mathfrak A^*\) is a completely continuous operator, where \(\theta_{\phi}\) is defined by \(\theta_{\phi}(a)=a \cdot \phi\) for all \(a \in \mathfrak A\). We call \(\mathfrak A\), a completely continuous Banach algebra if \(c.c(\mathfrak A)=\mathfrak A^*\). We give some examples of completely continuous Banach algebras and a sufficient condition for an open problem raised for the first time by \textit{J. E. Galé} et al. [Trans. Am. Math. Soc. 331, No. 2, 815--824 (1992; Zbl 0761.46037)]: does there exist an infinite dimensional amenable Banach algebra whose underlying Banach space is reflexive? We prove that a reflexive, amenable, completely continuous Banach algebra with the approximation property is trivial.A mathematical presentation of Laurent Schwartz's distributions.https://zbmath.org/1449.460282021-01-08T12:24:00+00:00"Alvarez, Josefina"https://zbmath.org/authors/?q=ai:alvarez-alonso.josefinaSummary: This is a mathematical presentation of Laurent Schwartz's distributions.Bochner-Riesz operators on weak Musielak-Orlicz Hardy spaces.https://zbmath.org/1449.420392021-01-08T12:24:00+00:00"Wang, Aiting"https://zbmath.org/authors/?q=ai:wang.aiting"Liu, Xiong"https://zbmath.org/authors/?q=ai:liu.xiong"Wang, Wenhua"https://zbmath.org/authors/?q=ai:wang.wenhua"Li, Baode"https://zbmath.org/authors/?q=ai:li.baodeSummary: Let \(\varphi:\mathbb{R}^n \times [0, \infty) \to [0, \infty)\) satisfy that \(\varphi (x, \cdot)\), for any given \(x \in \mathbb{R}^n\), is an Orlicz function and \(\varphi (\cdot, t)\) is a Muckenhoupt \({A_\infty}\) weight uniformly in \(t \in (0, \infty)\). In this paper, by using the atomic decomposition of the weak Musielak-Orlicz Hardy space \({\mathrm{WH}}^\varphi (\mathbb{R}^n)\) and a subtle pointwise estimate for the non-tangential grand maximal function of the Bochner-Riesz operators \(T_R^\delta\), we obtain that \(T_R^\delta\) is bounded on \({\mathrm{WH}}^\varphi (\mathbb{R}^n)\). The result is also new even when \(\varphi (x, t):= \Phi (t)\) for all \( (x, t) \in \mathbb{R}^n\times [0, \infty)\), where \(\Phi\) is an Orlicz function.On solvability of inhomogeneous boundary-value problems in Sobolev-Slobodetskiy spaces.https://zbmath.org/1449.471332021-01-08T12:24:00+00:00"Mikhailets, V. A."https://zbmath.org/authors/?q=ai:mikhailets.vladimir-a|mikhailets.volodymyr-a"Skorobohach, T. B."https://zbmath.org/authors/?q=ai:skorobohach.t-bSummary: We investigate the most general class of Fredholm one-dimensional boundary-value problems in the Sobolev-Slobodetskiy spaces. Boundary conditions of these problems may contain a derivative of the whole or fractional order. It is established that each of these boundary-value problems corresponds to a certain rectangular numerical characteristic matrix with kernel and cokernel having the same dimension as the kernel and cokernel of the boundary-value problem. Sufficient conditions for the sequence of the characteristic matrices of a specified boundary-value problems to converge are found.Localization property for the convolution of generalized periodic functions.https://zbmath.org/1449.460292021-01-08T12:24:00+00:00"Gorodets'kyĭ, V. V."https://zbmath.org/authors/?q=ai:gorodetskyi.vasyl-v|gorodetskij.v-v"Martynyuk, O. V."https://zbmath.org/authors/?q=ai:martynyuk.olga-vSummary: The well-known Riemann localization principle for the Fourier series of summable functions is reformulated for the convolution of generalized periodic functions with families of functions, which usually coincide with kernels of certain linear methods of summation of Fourier series (for example, summation methods such as the Gauss-Weierstrass one). We call the families of functions, for which the Riemann localization holds, the families of functions of a class \(L(X)\). The necessary and sufficient conditions of belonging the family of functions to the class \(L(X)\) are found in the case where \(X\) is a sufficiently broad non-quasi-analytic class of periodic functions or \(X\) is a class of analytic periodic functions (in particular, \(X =G_{\{\beta\}}\) for \(\beta > 1\) and \(X =G_{\{\beta\}}\) if \(0 <\beta\leq1\)). The definition of ``analytic functional equal to zero on an open set'' is also substantiated; a specific example of analytic functional is given, which is 0 on \((a, b)\subset[0, 2\pi]\). The use of the obtained result in partial differential equation theory allows us to obtain a new property (localization property, the property of local convergence improvement) of many problems of mathematical physics, since such solutions are often depicted as a convolution of some family of basic functions from the space \(X\) with a function \(F\) defined at the boundary of the domain, \(F\) may be a generalized function from a space \(X'\).Applications of four dimensional matrices to generating some roughly $I_2$-convergent double sequence spaces.https://zbmath.org/1449.460082021-01-08T12:24:00+00:00"Raj, Kuldip"https://zbmath.org/authors/?q=ai:raj.kuldip"Sharma, Charu"https://zbmath.org/authors/?q=ai:sharma.charuSummary: \textit{E. Dündar} [Numer. Funct. Anal. Optim. 37, No. 4, 480--491 (2016; Zbl 1365.40004)] introduced the notion of rough $I_2$-convergence and obtained two criteria for rough $I_2$-convergence. In the present paper we introduce some new rough ideal convergent sequence spaces of Musielak-Orlicz functions and four dimensional bounded-regular matrices. We study some topological and algebraic properties of these spaces. We also establish some inclusion relations between these spaces. Finally, we examine that these spaces are normal as well as monotone and sequence algebras.Isomorphic embeddings of Banach spaces.https://zbmath.org/1449.460112021-01-08T12:24:00+00:00"Zheng, Bentuo"https://zbmath.org/authors/?q=ai:zheng.bentuoSummary: This paper reviewed the studies on isomorphic embeddings of Banach spaces into superspaces with Schauder bases, shrinking bases, bounded bases, unconditional bases and spreading bases. Major problems and related achievements are presented in the hope that such a survey would be helpful to researches on this topic.The generalized Jordan-von Neumann type constant and normal structure.https://zbmath.org/1449.460202021-01-08T12:24:00+00:00"Zuo, Zhanfei"https://zbmath.org/authors/?q=ai:zuo.zhanfeiSummary: The generalized Jordan-von Neumann type constant is introduced, some basic properties of this new constant are investigated. Moreover, the weakly convergent sequence coefficient WCS\( (X)\) of Banach spaces is estimated by the generalized Jordan-von Neumann type constant, the weak orthogonality coefficient \(\mu (X)\) and Domínguez-Benavides coefficient \(R (1, X)\), which enables us to obtain some sufficient conditions for normal structure. The results obtained in this paper significantly improve some known results in the literatures.Piecewise-polynomial signal segmentation using convex optimization.https://zbmath.org/1449.940412021-01-08T12:24:00+00:00"Rajmic, Pavel"https://zbmath.org/authors/?q=ai:rajmic.pavel"Novosadová, Michaela"https://zbmath.org/authors/?q=ai:novosadova.michaela"Daňková, Marie"https://zbmath.org/authors/?q=ai:dankova.marieSummary: A method is presented for segmenting one-dimensional signal whose independent segments are modeled as polynomials, and which is corrupted by additive noise. The method is based on sparse modeling, the main part is formulated as a convex optimization problem and is solved by a proximal splitting algorithm. We perform experiments on simulated and real data and show that the method is capable of reliably finding breakpoints in the signal, but requires careful tuning of the regularization parameters and internal parameters. Finally, potential extensions are discussed.A note on weak 2-local isometries on differentiable function spaces.https://zbmath.org/1449.460142021-01-08T12:24:00+00:00"Li, Lei"https://zbmath.org/authors/?q=ai:li.lei.1|li.lei|li.lei.2|li.lei.5|li.lei.3|li.lei.7|li.lei.6|li.lei.4Summary: Let \({C^1}[0,1]\) be the Banach algebra of complex valued continuously differentiable functions with the norm \({\|f\|_L} = \max\{|f (x)| +|f' (x)|:x \in [0,1]\}\) or \({\|f\|_s} = {\|f\|_\infty} + {\|{f'}\|_\infty}\). The weak-2-local isometries in \({C^1}[0,1]\) are linear maps.\(p\)-adic BMO and VMO functions.https://zbmath.org/1449.110732021-01-08T12:24:00+00:00"Zelenov, Evgeniĭ Igor'evich"https://zbmath.org/authors/?q=ai:zelenov.evgenii-iSummary: Spaces of \(p\)-adic BMO and VMO functions are considered. It is proved that locally constant functions are dense in VMO space under BMO norm.Generalized functions asymptotically homogeneous along the unstable degenerated node.https://zbmath.org/1449.460322021-01-08T12:24:00+00:00"Drozhzhinov, Yuriĭ Nikolaevich"https://zbmath.org/authors/?q=ai:drozhzhinov.yu-n"Zav'yalov, Boris Ivanovich"https://zbmath.org/authors/?q=ai:zavialov.boris-ivanovichSummary: The generalized functions which have quasiasymptotics along the trajectories of one-parametric group are called asymptomatically homogeneous. The corresponding limit functions are homogeneous with respect to this group. In this paper we give the full description of asymptotically homogeneous generalized functions along the trajectories of unstable degenerated node. The obtained results are applied for description of homogeneous generalized functions for such trajectories in two dimensional case.Hardy type unique continuation properties for abstract Schrödinger equations and applications.https://zbmath.org/1449.353632021-01-08T12:24:00+00:00"Shakhmurov, Veli"https://zbmath.org/authors/?q=ai:shakhmurov.veli-bSummary: In this paper, Hardy's uncertainty principle and unique continuation properties of Schrödinger equations with operator potentials in Hilbert space-valued \(L^{2}\) classes are obtained. Since the Hilbert space \(H\) and linear operators are arbitrary, by choosing the appropriate spaces and operators we obtain numerous classes of Schrödinger type equations and its finite and infinite many systems which occur in a wide variety of physical systems.Amenability and Fan-Glicksberg theorem for set-valued mappings.https://zbmath.org/1449.430012021-01-08T12:24:00+00:00"Lau, Anthony To-Ming"https://zbmath.org/authors/?q=ai:lau.anthony-to-ming"Yao, Liangjin"https://zbmath.org/authors/?q=ai:yao.liangjinSummary: In this paper, we begin by discussion of some well known results on the existence of left invariant means in the spaces: \(LUC(S), AP(S)\) and \(WAP(S)\) with Hahn-Banach extension theorem. We then give a new and precise proof of the well known Fan-Glicksberg fixed point theorem. This is then followed by a discussion on some related open problems.\(C^1\)-smooth dependence on initial conditions and delay: spaces of initial histories of Sobolev type, and differentiability of translation in \(L^p\).https://zbmath.org/1449.342152021-01-08T12:24:00+00:00"Nishiguchi, Junya"https://zbmath.org/authors/?q=ai:nishiguchi.junyaSummary: The objective of this paper is to clarify the relationship between the \(C^1\)-smooth dependence of solutions to delay differential equations (DDEs) on initial histories (i.e., initial conditions) and delay parameters. For this purpose, we consider a class of DDEs which include a constant discrete delay. The problem of \(C^1\)-smooth dependence is fundamental from the viewpoint of the theory of differential equations. However, the above mentioned relationship is not obvious because the corresponding functional differential equations have the less regularity with respect to the delay parameter. In this paper, we prove that the \(C^1\)-smooth dependence on initial histories and delay holds by adopting spaces of initial histories of Sobolev type, where the differentiability of translation in \(L^p\) plays an important role.On certain functional equations related to Jordan *-derivations in semiprime *-rings and standard operator algebras.https://zbmath.org/1449.160392021-01-08T12:24:00+00:00"Ashraf, Mohammad"https://zbmath.org/authors/?q=ai:ashraf.mohammad"Wani, Bilal Ahmad"https://zbmath.org/authors/?q=ai:wani.bilal-ahmadSummary: The purpose of this paper is to investigate identities with Jordan *-derivations in semiprime *-rings. Let \(R\) be a 2-torsion free semiprime *-ring. In this paper it is shown that, if \(R\) admits an additive mapping \(D : R \mapsto R\) satisfying either \(D(xyx) = D(xy)x^* + xyD(x)\) for all \(x,y\in R\), or \(D(xyx) = D(x)y^* x^* + xD(yx)\) for all pairs \(x,y \in R\), then \(D\) is a *-derivation. Moreover, this result makes it possible to prove that if \(R\) satisfies \(2D(x n) = D(x^{n-1})x^* + x^{n-1} D(x) + D(x)(x^*)^{n-1} + xD(x^{n-1})\) for all \(x\in R\) and some fixed integer \(n\geq 2\), then \(D\) is a Jordan *-derivation under some torsion restrictions. Finally, we apply these purely ring theoretic results to standard operator algebras \(\mathcal{A(H)}\). In particular, we prove that if \(\mathcal{H}\) is a real or complex Hilbert space, with \(\dim(\mathcal{H}) > 1\), admitting a linear mapping \(D : \mathcal{A(H)}\mapsto\mathcal{B(H)}\) (where \(\mathcal{B(H)}\) stands for the bounded linear operators) such that \[2D(A^n) = D(A^{n-1})A^* + A^{n-1} D(A) + D(A)(A^*)^{n-1} + AD(A^{n-1})\] for all \(A \in\mathcal{A(H)}\), then \(D\) is Jordan *-derivation.On some properties of relative capacity and thinness in weighted variable exponent Sobolev spaces.https://zbmath.org/1449.320122021-01-08T12:24:00+00:00"Unal, C."https://zbmath.org/authors/?q=ai:unal.cihan|unal.cemal"Aydin, I."https://zbmath.org/authors/?q=ai:aydin.ilknur|aydin.ismailLet \(p:\mathbb{R}^n\longrightarrow[1,+\infty)\) be a measurable function and let \(\vartheta:\mathbb{R}^n\longrightarrow(0,+\infty)\) be locally integrable. Denote by \(L^p_\vartheta(\mathbb{R}^n)\) the space of all measurable functions \(f\) such that \(\int_{\mathbb{R}^n}|f(x)|^{p(x)}\vartheta(x)dx<+\infty\) and let \(W^{1,p}_\vartheta(\mathbb{R}^n):=\{f\in L^p_\vartheta(\mathbb{R}^n): \partial{f}/\partial{x_j}\in L^p_\vartheta(\mathbb{R}^n),\;j=1,\dots,n\}\). The authors study the space \(W^{1,p}_\vartheta(\mathbb{R}^n)\) and various capacities associated with this space.
Reviewer: Marek Jarnicki (Kraków)Norm attaining Arens extensions on \(\ell_1\).https://zbmath.org/1449.460352021-01-08T12:24:00+00:00"Falcó, Javier"https://zbmath.org/authors/?q=ai:falco.javier"García, Domingo"https://zbmath.org/authors/?q=ai:garcia.domingo"Maestre, Manuel"https://zbmath.org/authors/?q=ai:maestre.manuel"Rueda, Pilar"https://zbmath.org/authors/?q=ai:rueda.pilarSummary: We study norm attaining properties of the Arens extensions of multilinear forms defined on Banach spaces. Among other related results, we construct a multilinear form on \(\ell_1\) with the property that only some fixed Arens extensions determined a priori attain their norms. We also study when multilinear forms can be approximated by ones with the property that only some of their Arens extensions attain their norms.Approximation properties and frames for Banach and operator spaces.https://zbmath.org/1449.460222021-01-08T12:24:00+00:00"Hu, Qianfeng"https://zbmath.org/authors/?q=ai:hu.qianfeng"Liu, Rui"https://zbmath.org/authors/?q=ai:liu.ruiSummary: We made a survey of the main results on approximation properties and frames for Banach spaces and operator spaces. By introducing Schauder frame and completely bounded frame, we presented the equivalent characterizations of the bounded approximation properties for Banach spaces and the completely bounded approximation properties for operator spaces. We also provided some examples and the duality theory on complemented embedding, and introduced some open questions.A note on farthest point problem in Banach spaces.https://zbmath.org/1449.460192021-01-08T12:24:00+00:00"Som, Sumit"https://zbmath.org/authors/?q=ai:som.sumit"Savas, Ekrem"https://zbmath.org/authors/?q=ai:savas.ekremSummary: Farthest point problem states that ``Must every uniquely remotal set in a Banach space be singleton?'' In this paper we introduce the notion of partial ideal statistical continuity of a function which is way weaker than continuity of a function. We give an example to show that partial ideal statistical continuity is weaker than continuity. In this paper we use ideal summability to give some answers to FPP problem which improves some former results. We prove that if \(E\) is a non-empty, bounded, uniquely remotal subset in a real Banach space \(X\) such that \(E\) has a Chebyshev center \(c\) and the farthest point map \(F:X\rightarrow E\) restricted to \([c,F(c)]\) is partially ideal statistically continuous at \(c\) then \(E\) is singleton.Functorial properties of \(\mathrm{Ext}_{u}(\cdot, \mathcal{B})\) when \(\mathcal{B}\) is simple with continuous scale.https://zbmath.org/1449.460622021-01-08T12:24:00+00:00"Ng, P. W."https://zbmath.org/authors/?q=ai:ng.ping-wong"Robin, Tracy"https://zbmath.org/authors/?q=ai:robin.tracyTo give a sample of the results proved in this paper we mention the following one. The authors show that \(\mathrm{Ext}_{u}(\mathcal{A}, \mathcal{B})\) is always an abelian group whenever the \(C^*\)-algebra \(\mathcal{A}\) is separable and nuclear and \(\mathcal{B}\) is a simple continuous scale \(C^*\)-algebra. A~similar result is proved in the unital case.
Reviewer: Cătălin Badea (Villeneuve d'Ascq)Best approximation and characterization of Hilbert spaces.https://zbmath.org/1449.410272021-01-08T12:24:00+00:00"Rajabi, Setareh"https://zbmath.org/authors/?q=ai:rajabi.setarehSummary: It is well known that for any nonempty closed convex subset \(C\) of a Hilbert space, any best approximation \(y\in C\) of the point \(x\) satisfies the inequality \(\Vert x-y\Vert^{2}+\Vert z-y\Vert^{2} \leq \Vert x-z\Vert^{2}\) for all \(z\in C\). In this paper, we first introduce and study a new subset of best approximations involving this inequality in general metric spaces. Then, we provide some equivalent conditions which characterize Hilbert spaces.The structure of integral representations in topological vector spaces.https://zbmath.org/1449.460272021-01-08T12:24:00+00:00"Meziani, Lakhdar"https://zbmath.org/authors/?q=ai:meziani.lakhdarSummary: The subject of the present work deals with integral representations of bounded operators, acting on linear spaces of vector valued functions.
First we will consider operators on the space \(C_0(S,X)\) of all continuous functions \(f:S\to X\) vanishing at infinity, endowed with the uniform topology, \(S\) being a locally compact space and \(X\) a Banach space. We will give a complete characterization of operators \(T:C_0(S,X) \to X\) which enjoy an integral form with respect to a scalar measure \(\mu\) on \(S\).
Next we consider integral representations for operators on \(L_1\) type spaces, with values in a Banach space or a locally convex space. The main setting is the Bochner integration process with respect to finite abstract measure. The integral representations obtained may be considered as generalizations of the classical Riesz Theorem.Supporting vectors of continuous linear projections.https://zbmath.org/1449.460122021-01-08T12:24:00+00:00"García-Pacheco, Francisco Javier"https://zbmath.org/authors/?q=ai:garcia-pacheco.francisco-javier"Naranjo-Guerra, Enrique"https://zbmath.org/authors/?q=ai:naranjo-guerra.enriqueSummary: This paper is aimed at studying the supporting vectors of continuous linear projections on Banach spaces, that is, the unit vectors at which a projection attains its norm. We first characterize supporting vectors of general continuous linear operators between Banach spaces. Then we characterize \(M\)-projections in terms of supporting vectors. Finally, by means of the supporting vectors again, we provide a characterization of projections of norm 1 on strictly convex Banach spaces.Pro-\(C^*\)-algebras associated with pro-\(C^*\)-correspondences versus tensor products.https://zbmath.org/1449.460452021-01-08T12:24:00+00:00"Joiţa, M."https://zbmath.org/authors/?q=ai:joita.mariaA complete Hausdorff topological \(*\)-algebra whose topology is defined by a directed family of \(C^{*}\)-seminorms is called a pro-\(C^{*}\)-algebra.
The author proves that, under natural conditions, the minimal, respectively maximal tensor product of a pro-\(C^{*}\)-algebra associated to a pro-\(C^{*}\)--correspondence and a pro-\(C^{*}\)-algebra is isomorphic to the pro-\(C^{*}\)-algebra associated with a tensor product pro-\(C^{*}\)-correspondence.
Reviewer: Ömer Gök (Istanbul)The dual pairs of the left and right limit spaces for Bochner integrable space \({L^p} (\mu, X)\).https://zbmath.org/1449.460152021-01-08T12:24:00+00:00"Yang, Peikang"https://zbmath.org/authors/?q=ai:yang.peikang"Luo, Cheng"https://zbmath.org/authors/?q=ai:luo.chengSummary: The dual of Bochner integrable space \({L^p} (\mu, X)\) depends on the property of Banach space \(X\), therefore, we focus on the Banach space \(X\) of which the dual space has Radon-Nikodym property. Applying the conclusions of a literature, we obtain the dual spaces of the the left limit and right limit spaces of \({L^p} (\mu, X)\). Then we obtain that if \(X\) is reflexive, then \({L^{p-0}} (\mu, X)\), \({L^{q + 0}} (\mu, X)$ ($1 < p, q < \infty$), \({L^{\infty - 0}} (\mu, X)\) and \({L^{1 + 0}} (\mu, X)\) are also reflexive. Finally we obtain the dual pair of the left and right spaces, and the definition of polar topologies and some properties for \({L^p} (\mu, X)\).Boundedness of self-map composition operators for two types of weights on the upper half-plane.https://zbmath.org/1449.470502021-01-08T12:24:00+00:00"Ardalani, Mohammad Ali"https://zbmath.org/authors/?q=ai:ardalani.mohammad-aliSummary: In this paper we find conditions for boundedness of self-map composition operators on weighted spaces of holomorphic functions on the upper half-plane for two kinds of weights which are of moderate growth.Martingale transforms and weak Hardy-Orlicz-Karamata spaces of martingales.https://zbmath.org/1449.600842021-01-08T12:24:00+00:00"Zhou, Nian"https://zbmath.org/authors/?q=ai:zhou.nian"Yu, Lin"https://zbmath.org/authors/?q=ai:yu.linSummary: The relations between weak Hardy-Orlicz-Karamata spaces of martingales are characterized by martingale transforms. More precisely, let \({\Phi_1}\underline \prec {\Phi_2}\) be two Young functions and \(b (\cdot)\) be non-decreasing slowly varying function, it is constructively proved that the elements in the weak Hardy-Orlicz-Karamata space \(w{\mathcal{H}_{\Phi_{1,}b}}\) are none other than the martingale transforms of those in weak Hardy-Orlicz-Karamata space \(w{\mathcal{H}_{\Phi_{2,}b}}\). The results obtained here extend the corresponding results in former literature.On a nonlinear Peetre's theorem in full Colombeau algebras.https://zbmath.org/1449.460332021-01-08T12:24:00+00:00"Nigsch, E. A."https://zbmath.org/authors/?q=ai:nigsch.eduard-albertIn previous works the author has supplied the functional analytic foundations for diffeomorphism invariant Colombeau-type algebras in the so-called full setting, where canonical embeddings of spaces of distributions are available. Representatives of such generalized functions are smooth maps (in the sense of convenient calculus) of the form \(R:C^\infty(\Omega,\mathcal{D}(\Omega))\to C^\infty(\Omega)\). Here, \(\Omega\) is an open set and \(\mathcal{D}(\Omega)\) is the space of test functions on \(\Omega\). To obtain sheaf properties in full Colombeau algebras, certain locality properties have to be imposed on \(R\). This paper builds on methods developed by \textit{J. Slovák} in [Ann. Global Anal. Geom. 6, No. 3, 273--283 (1988; Zbl 0636.58042)] to derive a non-linear Peetre theorem for representatives as above.
In particular, locality is characterized as \(R(\vec{\varphi})(x)\) depending only on the infinite order jet of \(\vec{\varphi}\) at \(x\) instead of the full germ of \(\vec{\varphi}\) at \(x\). In addition, it is shown that if \(R\) is smooth and local then for any \(\vec{\varphi}\) and \(x_0\), there exists a neighborhood of \((\vec{\varphi},x_0)\) and a natural number \(r\) such that for all \((\vec{\psi},x)\) in this neighborhood, \(R(\vec{\psi})(x)\) depends only on the \(r\)-jet of \(\vec{\psi}\) at~\(x\).
Reviewer: Michael Kunzinger (Wien)\(CM\)-ideals and \(L^1\)-matricial split faces.https://zbmath.org/1449.460132021-01-08T12:24:00+00:00"Ghatak, Anindya"https://zbmath.org/authors/?q=ai:ghatak.anindya"Karn, Anil Kumar"https://zbmath.org/authors/?q=ai:karn.anil-kumarThe paper contains three main new results (the notation is standard) presented in Section~1:
\textbf{Theorem 1.} Let \(V\) be an operator space and \(W\) a closed subspace. Then \(W\) is a CM-ideal in \(V\) exactly when \(W^\perp\) is a CL-summand in the matricial dual \(V^\ast\) of \(V\).
\textbf{Theorem 2.} Let \(V\) be a matricially order smooth \(\infty\)-normed space and \(W\) a closed self-adjoint subspace. Then \(W\) is a CM-ideal in \(V\) exactly when \(M_n(W)_{sa}\) is an M-ideal in \(M_n(V)\) for each \(n\in\mathbb{N}\).
\textbf{Theorem 3.} Let \(V\) be an a matricially order smooth \(\infty\)-normed space and \(W\) a self-adjoint subspace. Then \(W\) is a CM-ideal in \(V\) exactly when \(\{M_n(W^\perp)\cap Q_n(V)\}\) is an \(L^1\)-matricial split face of \(Q_n(V)\) for each \(n\in\mathbb{N}\).
Section 2 contains the needed preliminaries while Section~3 and Section~4 present proofs of Theorems~1 and~2. Theorem~3 is an application of Theorems~1 and~2, and is one of many applications presented in Sections~5 and~6.
Reviewer: Olav Nygaard (Kristiansand)Isometries on certain non-complete vector-valued function spaces.https://zbmath.org/1449.470712021-01-08T12:24:00+00:00"Mojahedi, Mojtaba"https://zbmath.org/authors/?q=ai:mojahedi.mojtaba"Sady, Fereshteh"https://zbmath.org/authors/?q=ai:sady.fereshtehFor compact Hausdorff spaces \(X,Y\) and Banach spaces \(E,F\), the authors consider dense linear subspaces \(A,B\) of \(C(X, E)\) and \(C(Y, F)\), respectively, equipped with norms \(\|\cdot\|_A = \max\left(\|\cdot\|_\infty, p(\cdot) \right)\), \(\|\cdot\|_B = \max\left(\|\cdot\|_\infty, q(\cdot) \right)\), where \(p\) and \(q\) are seminorms that vanish on constant functions. The paper is devoted to sufficient conditions on the spaces involved and on a bijective isometry \(T : A \to B\) ensure the existence of a representation of \(T\) of the form of weighted composition operator \[(Tf)(y) = V_y(f(\Phi(y)),\] where \(\Phi: Y \to X\) is a homeomorphism, and each \(V_y : E \to F\) is an isometry. The conditions on the spaces are of geometric nature and they are weaker than strict convexity. The results of the paper complement previously known resuls for isometries of Lipschitz spaces and of spaces of absolutely continuous vector-valued functions.
Reviewer: Vladimir Kadets (Kharkiv)Distributional boundary values of generalized Hardy functions in Beurling's tempered distributions.https://zbmath.org/1449.320072021-01-08T12:24:00+00:00"Sohn, Byung Keun"https://zbmath.org/authors/?q=ai:sohn.byung-keunLet \(C\) be an open convex cone in \(\mathbb{R}^N\) and let \(T^C=\mathbb{R}^N+iC\) in \(\mathbb{C}^N\). In this paper the author defines a generalization of Hardy functions (\(1 \leq p < \infty\)) on \(T^C\) and extended tempered distribution space \(S_{w}'\) of Beurling's tempered distribution space \(S_{(w)}'\) for a weight function \(w\). The author obtains the analytical and topological properties of \(S_{w}'\) and shows that the generalized Hardy functions (\(1< p \leq 2\)), have distributional boundary values in the weak topology of \(S_{(w)}'\) using the analytical properties of \(S_{w}'\) .
Reviewer: Koichi Saka (Akita)Relative position of three subspaces in a Hilbert space.https://zbmath.org/1449.460232021-01-08T12:24:00+00:00"Enomoto, Masatoshi"https://zbmath.org/authors/?q=ai:enomoto.masatoshi"Watatani, Yasuo"https://zbmath.org/authors/?q=ai:watatani.yasuoSummary: We study the relative position of three subspaces in an infinite dimensional Hilbert space. In the finite-dimensional case over an arbitrary field, \textit{S. Brenner} [J. Algebra 6, 100--114 (1967; Zbl 0229.16020)] described the general position of three subspaces completely. We extend it to a certain class of three subspaces in an infinite-dimensional Hilbert space over the complex numbers.Characterizations of certain Hankel transform involving Riemann-Liouville fractional derivatives.https://zbmath.org/1449.420092021-01-08T12:24:00+00:00"Upadhyay, S. K."https://zbmath.org/authors/?q=ai:upadhyay.santosh-kumar"Khatterwani, Komal"https://zbmath.org/authors/?q=ai:khatterwani.komalSummary: In this paper, the relation between the two dimensional fractional Fourier transform and the fractional Hankel transform is discussed in terms of radial functions. Various operational properties of the Hankel transform and the fractional Hankel transform are studied involving Riemann-Liouville fractional derivatives. The application of the fractional Hankel transform in networks with time varying parameters is given.The existence of fuzzy Dedekind completion of Archimedean fuzzy Riesz space.https://zbmath.org/1449.460632021-01-08T12:24:00+00:00"Iqbal, Mobashir"https://zbmath.org/authors/?q=ai:iqbal.mobashir"Bashir, Zia"https://zbmath.org/authors/?q=ai:bashir.ziaSummary: The fuzzy Riesz space is an attempt to study vector spaces with fuzzy ordering to model scenarios of more vague nature. The aim of this paper is to prove the existence of fuzzy Dedekind completion, whereas to achieve this goal, other related concepts like fuzzy order convergence, fuzzy positive operators, and their related results are also explored to enrich the theory of fuzzy Riesz spaces.Applications of Riesz mean and lacunary sequences to generate Banach spaces and AK-BK spaces.https://zbmath.org/1449.460072021-01-08T12:24:00+00:00"Raj, Kuldip"https://zbmath.org/authors/?q=ai:raj.kuldip"Esi, Ayhan"https://zbmath.org/authors/?q=ai:esi.ayhan"Pandoh, Suruchi"https://zbmath.org/authors/?q=ai:pandoh.suruchiSummary: In this paper we establish some wide-ranging spaces of sequences generated by Riesz mean associated with lacunary sequences and multiplier sequences of Orlicz function. We have encompassed some topological and algebraic properties of these sequence spaces. We also make an effort to prove that these spaces are Banach and AK-BK spaces. Finally, we prove that these sequence spaces are topologically isomorphic.A new difference sequence set of order \(\alpha\) and its geometrical properties.https://zbmath.org/1449.460042021-01-08T12:24:00+00:00"Et, Mikail"https://zbmath.org/authors/?q=ai:et.mikail"Karakaya, Vatan"https://zbmath.org/authors/?q=ai:karakaya.vatanSummary: We introduce a new class of sequences named as \(m_\alpha (\Delta^r, \phi, p)\) and, for this space, we study some inclusion relations, topological properties, and geometrical properties such as order continuous, the Fatou property, and the Banach-Saks property of type \(p\).Cesàro summable sequence spaces over the non-Newtonian complex field.https://zbmath.org/1449.460052021-01-08T12:24:00+00:00"Kadak, Uğur"https://zbmath.org/authors/?q=ai:kadak.ugurSummary: The spaces \(\omega_0^p\), \(\omega^p\), and \(\omega_{\infty}^p\) can be considered the sets of all sequences that are strongly summable to zero, strongly summable, and bounded, by the Cesàro method of order \(1\) with index \(p\). Here we define the sets of sequences which are related to strong Cesàro summability over the non-Newtonian complex field by using two generator functions. Also we determine the \(\beta\)-duals of the new spaces and characterize matrix transformations on them into the sets of \(\ast\)-bounded, \(\ast\)-convergent, and \(\ast\)-null sequences of non-Newtonian complex numbers.Erratum to: Atomic decomposition of Hardy spaces and characterization of \(BMO\) via Banach function spaces.https://zbmath.org/1449.460252021-01-08T12:24:00+00:00"Ho, K.-P."https://zbmath.org/authors/?q=ai:ho.keang-po|ho.kwok-punErratum to \textit{K.-P. Ho} [Anal. Math. 38, No. 3, 173--185 (2012; Zbl 1289.46049)].On a ``Martingale property'' of Franklin series.https://zbmath.org/1449.420482021-01-08T12:24:00+00:00"Gevorkyan, G. G."https://zbmath.org/authors/?q=ai:gevorkyan.gegham-gA nested sequence of partitions of the interval \([0,1]\) into \(n=2^\mu+\nu\) subintervals is defined by dividing \([0,1]\) into \(2^\mu\) intervals of length \(2^{-\mu}\) and then adding the midpoints of the first \(\nu\) of these intervals. Let \(S_n\) be the set of piecewise linear continuous functions on the subintervals. Then the Franklin system consists of the sequence of orthogonal functions \(\{f_n\}_{n=0}^\infty\) with \(f_n\in S_n\) orthogonal to \(S_{n-1}\) for \(n\ge2\). A function \(f\in L^2[0,1]\) can be expanded as \(f(x)=\sum_{n=0}^\infty a_nf_n(x)\) with partial sums \(\sigma_n(x)=\sum_{k=0}^n a_k f_k(x)\). This paper derives convergence results that are analogous to convergence results obtained for other systems such as the Haar system [\textit{F. G. Arutyunyan}, Dokl., Akad. Nauk Arm. SSR 42, 134--140 (1966; Zbl 0178.40802)] and martingales [\textit{R. F. Gundy}, Trans. Am. Math. Soc. 124, 228--248 (1966; Zbl 0158.35801)]. Although the Franklin system is not a martingale, it has similar properties, which are exploited here to obtain similar convergence results. The key result in this derivation is the theorem saying that \(\inf_n\sigma_n(x)>-\infty\) on \(E\) implies that \(\sup_n\sigma_n(x)<+\infty\) a.e. on \(E\).
Reviewer: Adhemar Bultheel (Leuven)On decompositions of continuous generalized frames in Hilbert spaces.https://zbmath.org/1449.420672021-01-08T12:24:00+00:00"Zhang, Wei"https://zbmath.org/authors/?q=ai:zhang.wei.10|zhang.wei.6|zhang.wei.16|zhang.wei.4|zhang.wei.12|zhang.wei.17|zhang.wei.18|zhang.wei.3|zhang.wei.11|zhang.wei.15|zhang.wei.5|zhang.wei.2|zhang.wei.13|zhang.wei.14|zhang.wei.7|zhang.wei.9"Fu, Yanling"https://zbmath.org/authors/?q=ai:fu.yanlingSummary: This paper establishes the characterization of continuous generalized frames, Parseval continuous \(g\)-frames, continuous generalized Riesz bases and continuous generalized orthonormal bases in terms of the continuous generalized preframe operator. Using the established characterization results and decompositions of bounded operators, the representation of continuous generalized frames in terms of linear combinations of simpler ones such as continuous generalized orthonormal bases, continuous generalized Riesz bases and Parseval continuous generalized frames is studied.A few observations on Weaver's quantum relations.https://zbmath.org/1449.460512021-01-08T12:24:00+00:00"González-Pérez, Adrián M."https://zbmath.org/authors/?q=ai:gonzalez-perez.adrian-mThe author proves that quantum relations over a von Neumann algebra \(M\) are in bijective correspondence with weakly closed left ideals in the extended Haagerup tensor product \(M \otimes_{eh} M\). Also, he studies invariant quantum relations over a group von Neumann algebra, as an application.
Reviewer: Ömer Gök (Istanbul)Duals of Hardy amalgam spaces and norm inequalities.https://zbmath.org/1449.420382021-01-08T12:24:00+00:00"Ablé, Z. V. P."https://zbmath.org/authors/?q=ai:able.z-v-p"Feuto, J."https://zbmath.org/authors/?q=ai:feuto.justinThere are many generalizations of the classical Hardy spaces by taking the norm of the maximal function in certain spaces rather than in the Lebesgue ones. The author's choice for replacing is the Wiener amalgam spaces. In the paper under review, they first study characterizations of such spaces, including the atomic ones. Then they characterize the dual spaces of the generalized Hardy spaces defined in the above way. Finally, they prove that in these generalized Hardy spaces some classical singular operators, such as Calderón-Zygmund, convolution and Riesz potential operators, are bounded.
Reviewer: Elijah Liflyand (Ramat-Gan)\((\alpha ,\beta )\)-\(A\)-normal operators in semi-Hilbertian spaces.https://zbmath.org/1449.470422021-01-08T12:24:00+00:00"Benali, Abelkader"https://zbmath.org/authors/?q=ai:benali.abelkader"Ould Ahmed Mahmoud, Sid Ahmed"https://zbmath.org/authors/?q=ai:sid-ahmed.ould-ahmed-mahmoudSummary: Let \({\mathcal{H}}\) be a Hilbert space and let \(A\) be a positive bounded operator on \({\mathcal{H}}\). The semi-inner product \(\langle u\,|\,v \rangle _A:=\langle Au\,|\,v\rangle\), \(u,v \in{\mathcal{H}}\), induces a semi-norm \(\left\| .\right\| _A\) on \({\mathcal{H}}.\) This makes \({\mathcal{H}}\) into a semi-Hilbertian space. In this paper, we introduce a new class of operators called \((\alpha ,\beta )\)-\(A\)-normal operators in semi-Hilbertian spaces. Some structural properties of this class of operators are established.Extension of factorization theorems of Maurey to \(s\)-positively homogeneous operators.https://zbmath.org/1449.470392021-01-08T12:24:00+00:00"Tiaiba, Abdelmoumen"https://zbmath.org/authors/?q=ai:tiaiba.abdelmoumenSummary: In the present work, we prove that the class of \(s\)-positively homogeneous operators is a Banach space. As application, we give the generalization of some Maurey factorization theorems to \(T\) which is a \(s\)-positively homogeneous operator from \(X\) a Banach space into \(L_p\). We establish necessary and sufficient conditions to prove that \(T\) factors through \(L_q\). After this we extend the dual factorization theorem to the same class of operators above.Phase transitions on \(C^*\)-algebras arising from number fields and the generalized Furstenberg conjecture.https://zbmath.org/1449.460572021-01-08T12:24:00+00:00"Laca, Marcelo"https://zbmath.org/authors/?q=ai:laca.marcelo"Warren, Jacqueline M."https://zbmath.org/authors/?q=ai:warren.jacqueline-mFor an algebraic number field \(K\) with ring of integers \(O_K\), the (multiplicative) monoid \(O_K^{\times}\) of non-zero integers action on the (additive) group \(O_K\) gives rise to the semi-direct product \(O_K\rtimes O_K^{\times}\), called here the ``affine'' or ``\(ax+b\)'' monoid of algebraic integers in \(K\). The Toeplitz-like \(C^*\)-algebra generated by the left regular representation of the \(ax+b\) monoid acting by isometries on \(\ell^2(O_K\rtimes O_K^{\times})\) was studied by \textit{J. Cuntz} et al. [Math. Ann. 355, No. 4, 1383--1423 (2013; Zbl 1273.22008)], who analysed the equilibrium states of the time evolution on this \(C^*\)-algebra determined by the absolute norm, and characterized the simplex of KMS equilibrium states of this dynamical system for any inverse temperature \(\beta\in(0,\infty]\).
In the paper under review, the low-temperature range of the classification of KMS equilibrium states is studied, using the parametrization in terms of tracial states of direct sums of group \(C^*\)-algebras. Because of the action of units arising here, a higher-dimensional version of Furstenberg's seminal conjecture on rigidity for probability measures on the circle invariant under the multiplicative action of a non-lacunary semigroup of integers [\textit{H. Furstenberg}, Math. Syst. Theory 1, 1--49 (1967; Zbl 0146.28502)] enters the picture. The main results classify the behaviours arising in terms of the ideal class group, the degree, and the unit rank of \(K\), and an explicit description of the primitive ideal space of the associated transformation group \(C^*\)-algebra for number fields of unit rank at least \(2\) that are not complex multiplication fields.
Reviewer: Thomas B. Ward (Leeds)Maps preserving Jordan multiple products on factor von Neumann algebras.https://zbmath.org/1449.460522021-01-08T12:24:00+00:00"Zhang, Fangjuan"https://zbmath.org/authors/?q=ai:zhang.fangjuan"Shi, Donghe"https://zbmath.org/authors/?q=ai:shi.donghe"Wang, Lihong"https://zbmath.org/authors/?q=ai:wang.lihongSummary: Based on the method of operator block, we obtained the characterization of maps preserving \({\mathrm{Jordan}}^*\) multiple products on factor von Neumann algebras. Let \(A\) and \(B\) be two factor von Neumann algebras and \({f_n} ({A_1}, {A_2}, \dots, {A_n}) = (f_{n-1} ({A_1}, {A_2}, \dots, {A_{n-1}}), {A_n})\) be \({\mathrm{Jordan}}^*\) multiple products of \({{A_1}, {A_2}, \dots, {A_n}}\). Let \(\varphi: A \to B\) be a bijective mapping, it satisfies \(\varphi ({f_n} ({A_1}, {A_2}, \dots, {A_n})) = {f_n} (\varphi ({A_1}), \varphi ({A_2}), \dots, \varphi ({A_n}))\) if and only if \(\varphi\) is a \(^*\)-ring isomorphism or a \(^*\)-ring anti-isomorphism.Topological algebras of locally solid vector subspaces of order bounded operators.https://zbmath.org/1449.460422021-01-08T12:24:00+00:00"Zabeti, Omid"https://zbmath.org/authors/?q=ai:zabeti.omidSummary: Let \(E\) be a locally solid vector lattice. In this paper, we consider two particular vector subspaces of the space of all order bounded operators on \(E\). With the aid of two appropriate topologies, we show that under some conditions, they establish both, locally solid vector lattices and topologically complete topological algebras.On \(G\)-Banach frames.https://zbmath.org/1449.420602021-01-08T12:24:00+00:00"Rathore, Ghanshyam Singh"https://zbmath.org/authors/?q=ai:singh.ghanshyam"Mittal, Tripti"https://zbmath.org/authors/?q=ai:mittal.triptiSummary: \textit{M. R. Abdollahpour} et al. [Methods Funct. Anal. Topol. 13, No. 3, 201--210 (2007; Zbl 1144.46010)] generalized the concepts of frames for Banach
spaces and defined \(g\)-Banach frames in Banach spaces. In the present paper, we
define various types of \(g\)-Banach frames in Banach spaces. Examples and counter
examples to distinguish various types of \(g\)-Banach frames in Banach spaces have
been given. It has been proved that if a Banach space \(\mathcal{X}\) has a Banach frame, then \(\mathcal{X}\) has a normalized tight \(g\)-Banach frame for \(\mathcal{X}\). A characterization of an exact \(g\)-Banach frame has been given. Also, we consider the finite sum of \(g\)-Banach frames and give a sufficient condition for the finite sum of \(g\)-Banach frames to be a \(g\)-Banach frame. Finally, a sufficient condition for the stability of \(g\)-Banach frames in Banach spaces which provides optimal frame bounds has been given.Nonlinear maps preserving mixed Lie triple \(\xi \)-product on factor von Neumann algebras.https://zbmath.org/1449.460532021-01-08T12:24:00+00:00"Zhou, You"https://zbmath.org/authors/?q=ai:zhou.you"Zhang, Jianhua"https://zbmath.org/authors/?q=ai:zhang.jianhuaSummary: In this paper, we prove that every bijective map preserving mixed Lie triple \(\xi\)-products with \(\xi \ne 1\) from a factor von Neumann algebra \(\mathcal{M}\) with dim \(\mathcal{M} > 1\) into another factor von Neumann algebra \(\mathcal{N}\) with dim \(\mathcal{N} > 1\) is of the form \(A \to \varepsilon \Psi (A)\), where \(\varepsilon \in \{1, -1\}\) and \(\Psi:\mathcal{M} \to \mathcal{N}\) is a linear or conjugate linear \(*\)-isomorphism when \(\xi \in \text bf{R}\) and \(\Psi\) is a linear \(*\)-isomorphism when \(\xi \in \text bf{C}\backslash \text bf{R}\).Continuity of tensor products of the Hilbert space.https://zbmath.org/1449.460022021-01-08T12:24:00+00:00"Xing, Meili"https://zbmath.org/authors/?q=ai:xing.meiliSummary: This paper introduces the concepts of direct systems and inductive limits of inner product spaces. It is proved that the inductive limit of the inner product spaces is unique in the sense of unitary equivalence. What's more, the inductive limit of the specific inner product space is constructed. Finally, it is proved that the tensor product of the Hilbert space is continuous with respect to the inductive limit.Diameter preserving maps on function spaces.https://zbmath.org/1449.460442021-01-08T12:24:00+00:00"Hosseini, Maliheh"https://zbmath.org/authors/?q=ai:hosseini.maliheh"Font, Juan J."https://zbmath.org/authors/?q=ai:font.juan-jThis paper deals with diameter-preserving mappings on function spaces. Let $X$ be a compact space and $A$ a subspace of $C(X)$. The subspace $A$ is a function space if $A$ separates points and contains the constant functions. The space of all constant functions is denoted by $C$. Let $A_d$ be the quotient space $A/C$ endowed with the diameter norm ${\|\cdot\|_d}$, where $\|\pi(f)\|_d $ is equal to the diameter of the range of $f$. The unit ball of the dual space $A_d^*$ is denoted by $B_{A^*_d}$ and the set of the extreme points of the unit ball is denoted by $\operatorname{ext}(B_{A^*_d})$. The authors consider the diametral boundary of $A$, denoted $dch(A)$ and defined as the set of pairs $(x_1,x_2) \in X\times X$ such that $\delta_{x_1,x_2} =\delta_{x_1} -\delta_{x_2}$ is in $\operatorname{ext}(B_{A^*_d})$ ($\delta_x$ denotes the point evaluation at $x$). They denote by $\tilde{X}$ the set of points $x$ in the Choquet boundary of $A$ such that there exists a point $x_1$ in the Choquet boundary of $A$ with $(x,x_1) \in dch(A)$. We now state verbatim the main result in the paper:
Theorem 3.1. Let $A_i$, $i=1,2$, be (complex or real-valued) function spaces on compact spaces $X_i$ satisfying the following conditions:
\begin{itemize}
\item[(1)]
the set $\{\delta_x: \, x \in X_i\}$ is linearly independent in $(A_i, \|\cdot \|_{\infty})^*$,
\item[(2)]
$dch(A_1)=\{(x,x_1): \, x,\, x_1 \in \tilde{X_1}, \,\, x \neq x_1\}$ .
\end{itemize}
Assume that $T :A_1 \to A_2$ is a surjective diameter-preserving map. Then there are a homeomorphism $\psi : \tilde{X_2} \to \tilde{X_1}$, a scalar $\lambda \in \mathbb{T}$ and a linear functional $L :A_1\to \mathbb{C}$ such that $Tf(y)=\lambda f(\psi(y))+L(f)$ for all $f\in A_1$ and $y \in \tilde{X_2}$.
Reviewer: Fernanda Botelho (Memphis) (MR3688934)Monotone coefficients in Orlicz function spaces equipped with the \(p\)-Amemiya norm.https://zbmath.org/1449.460172021-01-08T12:24:00+00:00"He, Xin"https://zbmath.org/authors/?q=ai:he.xin"Cui, Yun'an"https://zbmath.org/authors/?q=ai:cui.yunan"Ji, Dandan"https://zbmath.org/authors/?q=ai:ji.dandanSummary: The coefficients of monotonicity subject to Orlicz function space equipped with the \(p\)-Amemiya norm are calculated, as well as the upper (lower) local coefficients of monotonicity with respect to the points located on the unit sphere. In addition, the criteria are given for the space to possess weakly set-valued fixed point property.The second nonlinear mixed Lie triple derivations on factor von Neumann algebras.https://zbmath.org/1449.160872021-01-08T12:24:00+00:00"Zhou, You"https://zbmath.org/authors/?q=ai:zhou.you"Zhang, Jianhua"https://zbmath.org/authors/?q=ai:zhang.jianhuaSummary: Let \(\mathcal{M}\) be a factor von Neumann algebra with dim \(\mathcal{M} > 1\), and \(L:\mathcal{M} \to \mathcal{M}\) be the second nonlinear mixed Lie triple derivation, i.e., \(L\) satisfies \[L ([[A,B],C]_*) = [[L (A), B],C]_* + [[A, L (B)], C]_* + [[A, B], L (C)]_*\] for all \(A, B, C \in \mathcal{M}\). Then \(L\) is an additive \(*\)-derivation.Consistent operator semigroups and their interpolation.https://zbmath.org/1449.470802021-01-08T12:24:00+00:00"ter Elst, A. F. M."https://zbmath.org/authors/?q=ai:ter-elst.antonius-f-m"Rehberg, J."https://zbmath.org/authors/?q=ai:rehberg.joachimSummary: Under a mild regularity condition we prove that the generator of the interpolation of two \(C_0\)-semigroups is the interpolation of the two generators.Twisted topological graph algebras are twisted groupoid \(C^*\)-algebras.https://zbmath.org/1449.460462021-01-08T12:24:00+00:00"Kumjian, Alex"https://zbmath.org/authors/?q=ai:kumjian.alexander"Li, Hui"https://zbmath.org/authors/?q=ai:li.huiSummary: In \textit{H. Li} [Houston J. Math. 43, No. 2, 459--494 (2017; Zbl 1391.46067)], the second author showed how Katsura's construction of the \(C^*\)-algebra of a topological graph \(E\) may be twisted by a Hermitian line bundle \(L\) over the edge space \(E^1\). The correspondence defining the algebra is obtained as the completion of the compactly supported continuous sections of \(L\). We prove that the resulting \(C^*\)-algebra is isomorphic to a twisted groupoid \(C^*\)-algebra where the underlying groupoid is the Renault-Deaconu groupoid of the topological graph with Yeend's boundary path space as its unit space.Path connected components in the spaces of nonzero weighted composition operators with the strong operator topology. I.https://zbmath.org/1449.470532021-01-08T12:24:00+00:00"Izuchi, Kei Ji"https://zbmath.org/authors/?q=ai:izuchi.keiji"Izuchi, Yuko"https://zbmath.org/authors/?q=ai:izuchi.yukoSummary: It is determined path connected components in the space of weighted composition operators on \(H^{\infty}\) and the disk algebra with the strong operator topology.Generalized spectra of convolution operators.https://zbmath.org/1449.460412021-01-08T12:24:00+00:00"Kumar, G. Krishna"https://zbmath.org/authors/?q=ai:kumar.g-krishna"Kulkarni, S. H."https://zbmath.org/authors/?q=ai:kulkarni.s-hSummary: The article introduces an algebra of integral operators namely convolution operators. The spectra, pseudospectra and condition spectra of convolution operators are described using Banach algebra techniques and the results developed are illustrated with examples and figures.Completions of quantum group algebras in certain norms and operators which commute with module actions.https://zbmath.org/1449.460602021-01-08T12:24:00+00:00"Nemati, Mehdi"https://zbmath.org/authors/?q=ai:nemati.mehdiSummary: Let \(L^1_{\text{cb}}(\mathbb{G})\) (respectively \(L^1_{\text{M}}(\mathbb{G})\)) denote the closure of the quantum group algebra \(L^1(\mathbb{G})\) of a locally compact quantum group \(\mathbb{G}\), in the space of completely bounded (respectively bounded) double centralizers of \(L^1(\mathbb{G}\)). In this paper, we study quantum group generalizations of various results from Fourier algebras of locally compact groups. In particular, left invariant means on \(L^1_{\text{cb}}(\mathbb{G})^*\) and on \(L^1_{\text{M}}(\mathbb{G})^*\) are defined and studied. We prove that the set of left invariant means on \(L^\infty(\mathbb{G})\) and on \(L^1_{\text{cb}}(\mathbb{G})^*(L^1_{\text{M}}(\mathbb{G})^*\)) have the same cardinality. We also study the left uniformly continuous functionals associated with these algebras. Finally, for a Banach \(\mathcal{A}\)-bimodule \(\mathcal{X}\) of a Banach algebra \(\mathcal{A}\) we establish a contractive and injective representation from the dual of a left introverted subspace of \(\mathcal{A}^*\) into \(B_\mathcal{A}(\mathcal{X}^*)\). As an application of this result we show that if the induced representation \(\varPhi:L\mathcal{U}C_{\text{cb}}(\mathbb{G})^*\to B_{L^1_{\text{cb}}(\mathbb{G})}(L^\infty(\mathbb{G}))\) is surjective, then \(L^1_{\text{cb}}(\mathbb{G})\) has a bounded approximate identity. We also obtain a characterization of co-amenable quantum groups in terms of representations of quantum measure algebras \(M(\mathbb{G})\).Automatic continuity of almost conjugate Jordan homomorphism on Fréchet \(Q\)-algebras.https://zbmath.org/1449.460432021-01-08T12:24:00+00:00"Omidi, Mohammad Reza"https://zbmath.org/authors/?q=ai:omidi.mohammad-rezaSummary: In this paper, the notation of almost conjugate Jordan homomorphism between Fréchet algebras is introduced. It is proven that, under special hypotheses, every almost conjugate Jordan homomorphism on Fréchet algebras is an (weakly) almost homomorphism. Also, the automatic continuity of them is generalized.Some notions of \((\sigma, \tau)\)-amenability for unitization of Banach algebras.https://zbmath.org/1449.460402021-01-08T12:24:00+00:00"Ghaderi, Eghbal"https://zbmath.org/authors/?q=ai:ghaderi.eghbal"Naseri, Saber"https://zbmath.org/authors/?q=ai:naseri.saberSummary: Let \(\mathcal A\) be a Banach algebra and \(\sigma\) and \(\tau\) be continuous endomorphisms on \(\mathcal A\). In this paper, we investigate \((\sigma, \tau)\)-amenability and \((\sigma, \tau)\)-weak amenability for unitization of Banach algebras, and also the relation between of them. We introduce and study the concepts \((\sigma, \tau)\)-trace extention property, \((\sigma, \tau)\)-\(I\)-weak amenability and \((\sigma, \tau)\)-ideal amenability for \(\mathcal A\) and its unitization, where \(I\) is a closed two-sided ideal in \(\mathcal A\).Hvala's theorem for generalized left \(\sigma \)-derivation on \({C^*}\)-algebras.https://zbmath.org/1449.460592021-01-08T12:24:00+00:00"Jayalakshmi, K."https://zbmath.org/authors/?q=ai:jayalakshmi.k"Bharathi, M. V. L."https://zbmath.org/authors/?q=ai:bharathi.m-v-lSummary: Generalized left \(\sigma \)-derivations in noncommutative prime \({C^*}\)-algebras are classified. If \({f_1}\), \({f_2}\) are two generalized left \(\sigma \)-derivations of a 2-torsion free prime \({C^*}\)-algebra \(A\), then the product \({f_1}{f_2}\) is again generalized left \(\sigma \)-derivation if and only if one of the following possibilities holds: there exists \(\tau \in C\) such that either \({f_1} (x) = \sigma (x)\tau \) or \({f_2} (x) = \sigma (x)\tau \), there exists \(m\), \(n\) in \({Q_l} ({R_c})\) such that \({f_1} (x) = m\sigma (x)\) and \({f_2} (x) = n\sigma (x)\), there exists \(m\), \(n\) in \({Q_l} ({R_c})\) such that \({f_1} (x) = \sigma (x)m\) and \({f_2} (x) = \sigma (x)n\), there exists \(m\), \(n\) in \({Q_l} ({R_c})\) and \(\lambda, \mu \in C\) such that \({f_1} (x) = \sigma (x)m + n\sigma (x)\) and \({f_2} (x) = \sigma (x)\lambda + (\sigma (x)m - n\sigma (x))\mu\). Further if \(A\) is a noncommutative 2-torsion free \({C^*}\)-algebra and \({f_1}{f_2}:A \to \mathcal{M}\) is nonzero generalized left \(\sigma \)-derivations satisfying \([{f_1} (x), {f_2} (x)] = 0\) for all \(x \in A\), then there exists \(\lambda \in C\) such that \({f_1} (x) = \lambda{f_2} (x)$, $x \in A\). Also \(f = 0\) where \(f:A \to \mathcal{M}\) provided that \(f (x)^n = 0\) for \(n > 1\) and \(A\) is \(2,\dots, n-1\)-torsion free using GPI ring theory.Some results on almost Banach-Saks operators.https://zbmath.org/1449.470762021-01-08T12:24:00+00:00"Hafidi, M."https://zbmath.org/authors/?q=ai:hafidi.m"H'Michane, J."https://zbmath.org/authors/?q=ai:hmichane.jawad"Sarih, M."https://zbmath.org/authors/?q=ai:sarih.maati|sarih.mustaphaAn operator \(T:\ E\to X\) from a Banach lattice \(E\) to a Banach space \(X\) is said to be almost Banach-Saks if, for each bounded disjoint sequence \((x_n)\) in \(E\), \((Tx_n)\) has a subsequence whose Cesàro means are norm convergent in \(X\). The authors characterize those Banach lattices for which each operator is almost Banach-Saks. The relationship with other classes of operators is studied.
Reviewer: Anatoly N. Kochubei (Kyïv)The parallel sum for adjointable operators on Hilbert \({C^*}\)-modules.https://zbmath.org/1449.460492021-01-08T12:24:00+00:00"Luo, Wei"https://zbmath.org/authors/?q=ai:luo.wei"Song, Chuanning"https://zbmath.org/authors/?q=ai:song.chuanning"Xu, Qingxiang"https://zbmath.org/authors/?q=ai:xu.qingxiangSummary: The parallel sum for adjointable operators on Hilbert \({C^*}\)-modules is introduced and studied. Some results known for matrices and bounded linear operators on Hilbert spaces are generalized to the case of adjointable operators on Hilbert \({C^*}\)-modules. It is shown that there exist a Hilbert \({C^*}\)-module \(H\) and two positive operators \(A, B \in \mathcal{L}(H )\) such that the operator equation \({A^{1/2}} = {({A + B} )^{1/2}}X\), \(X \in \mathcal{L}(H )\) has no solution, where \(\mathcal{L}(H )\) denotes the set of all adjointable operators on \(H\).Limited and Dunford-Pettis operators on Banach lattices.https://zbmath.org/1449.470752021-01-08T12:24:00+00:00"Bouras, Khalid"https://zbmath.org/authors/?q=ai:bouras.khalid"El Aloui, Abdennabi"https://zbmath.org/authors/?q=ai:el-aloui.abdennabi"Elbour, Aziz"https://zbmath.org/authors/?q=ai:elbour.azizAn operator \(T:\ X\to Y\) (\(X,Y\) are Banach spaces) is called a Dunford-Pettis operator if \(T\) carries weakly convergent sequences to norm convergent ones; \(T\) is called limited if \(T'\) carries weakly* convergent sequences in \(Y'\) to norm convergent sequences in \(X'\). The authors find conditions on a pair \(E,F\) of Banach lattices under which any positive Dunford-Pettis operator \(T:\ E\to F\) is limited. In particular, in this case the norm on \(E'\) is order continuous or \(\dim F<\infty\). Some sufficient conditions are also found.
Reviewer: Anatoly N. Kochubei (Kyïv)On the distributivity equation for uni-nullnorms.https://zbmath.org/1449.460642021-01-08T12:24:00+00:00"Wang, Ya-Ming"https://zbmath.org/authors/?q=ai:wang.yaming"Liu, Hua-Wen"https://zbmath.org/authors/?q=ai:liu.huawenSummary: A uni-nullnorm is a special case of 2-uninorms obtained by letting a uninorm and a nullnorm share the same underlying $t$-conorm. This paper is mainly devoted to solving the distributivity equation between uni-nullnorms with continuous Archimedean underlying $t$-norms and $t$-conorms and some binary operators, such as, continuous $t$-norms, continuous $t$-conorms, uninorms, and nullnorms. The new results differ from the previous ones about the distributivity in the class of 2-uninorms, which have not yet been fully characterized.The localization principle for formal Fourier series summarized by Gauss-Weierstrass method.https://zbmath.org/1449.352612021-01-08T12:24:00+00:00"Gorodets'kyĭ, V. V."https://zbmath.org/authors/?q=ai:gorodetskij.v-v|gorodetskyi.vasyl-v"Martynyuk, O. V."https://zbmath.org/authors/?q=ai:martynyuk.olga-vSummary: For formal series identifying linear continuous functionals given on the space of trigonometric polynomials and summarized by Gauss-Weierstrass methods, we prove an analog of the Riemann localization principle: if \(\{f_1, f_2\}\subset L_1 [0, 2\pi]\) are converge on the interval \( (a, b)\subset [0, 2\pi]\), then at each segment \([a + \varepsilon, b-\varepsilon]\subset (a, b)\) their difference of Fourier series uniformly converges to zero. Generally speaking, the principle of localization for Fourier series of \(2\pi\)-periodic generalized functions is not fulfilled. When studying various problems of mathematical physics and analysis, it is often used not the Fourier series itself, but the series summarized by one or another regular method, so it is natural to fulfill the principle of localization for such series. For example, the solution of the Dirichlet problem for the Laplace equation in a unit circle is represented by the Fourier series of the boundary function summarized by the Abel-Poisson method; the solution of the Cauchy periodic problem for the equation of thermal conductivity and the initial condition in the space of generalized periodic functions is treated as a formal Fourier series of the initial function summarized by the Gauss-Weierstrass method. The paper investigates multiple Fourier series of periodic hyperfunctions and ultradistributions.Pro-Banach reduced crossed products.https://zbmath.org/1449.460372021-01-08T12:24:00+00:00"Huang, Lizhong"https://zbmath.org/authors/?q=ai:huang.lizhongSummary: The definitions of pro-Banach algebra dynamical systems, pro-Banach crossed products and pro-Banach algebra reduced crossed products were introduced based on the concepts of pro-Banach algebras with a bounded approximate left identity, the local compact groups and the strong continuous actions of group on pro-Banach algebra. It was proved that the pro-Banach crossed product algebras and the pro-Banach reduced crossed product algebras associated with the given pro-Banach algebra dynamical systems are isomorphic when the group is compact.The iterates of positive linear operators with the set of constant functions as the fixed point set.https://zbmath.org/1449.470772021-01-08T12:24:00+00:00"Cătinaş, Teodora"https://zbmath.org/authors/?q=ai:catinas.teodora"Otrocol, Diana"https://zbmath.org/authors/?q=ai:otrocol.diana"Rus, Ioan A."https://zbmath.org/authors/?q=ai:rus.ioan-aSummary: Let \(\Omega \subset \mathbb{R}^r\), \(p \in \mathbb{N}^*\), be a nonempty subset and \(B(\Omega)\) be the Banach lattice of all bounded real functions on \(\Omega\), equipped with ``sup norm''. Let \(X \subset B(\Omega)\) be a linear sublattice of \(B(\Omega)\) and \(A:X \in X\) be a positive linear operator with the constant functions as the fixed point set. In this paper, using the weakly Picard operators technique, we study the iterates of the operator \(A\). Some relevant examples are also given.Certain properties of the sequence space \(\tilde \ell \left ({M,p,q} \right)\) of non-absolute type using four tuple band matrix \(B\left ({\tilde r,\tilde s,\tilde t,\tilde u} \right)\).https://zbmath.org/1449.460102021-01-08T12:24:00+00:00"Tripathy, Nilambar"https://zbmath.org/authors/?q=ai:tripathy.nilambar"Dutta, Salila"https://zbmath.org/authors/?q=ai:dutta.salilaSummary: In the present paper the sequence space \(\tilde l\left ({M, p, q} \right)\) of non-absolute type is introduced using an Orlicz function \(M\) along with a semi-norm \(q\) which is the domain of the generalized difference matrix \(B\left ({\tilde r, \tilde s, \tilde t, \tilde u} \right)\) and some topological as well as geometric properties of this space are obtained.Multipliers between range spaces of co-analytic Toeplitz operators.https://zbmath.org/1449.300902021-01-08T12:24:00+00:00"Fricain, Emmanuel"https://zbmath.org/authors/?q=ai:fricain.emmanuel"Hartmann, Andreas"https://zbmath.org/authors/?q=ai:hartmann.andreas"Ross, William T."https://zbmath.org/authors/?q=ai:ross.william-t-junSummary: In this paper we discuss the multipliers between range spaces of co-analytic Toeplitz operators.Characterizations of inner spaces under strongly \(E\)-convex set-valued mappings.https://zbmath.org/1449.470872021-01-08T12:24:00+00:00"Li, Ru"https://zbmath.org/authors/?q=ai:li.ru"Yu, Guolin"https://zbmath.org/authors/?q=ai:yu.guolin"Kong, Xiangyu"https://zbmath.org/authors/?q=ai:kong.xiangyuSummary: In this note, a kind of generalized strongly convex set-valued mappings, termed strongly \(E\)-convex set-valued mappings, is introduced in real normed spaces. Then, by employing Rådström cancellation law, some basic properties of strongly \(E\)-convex set-valued mappings are proposed. Finally, a characterization of inner product spaces involving the strongly \(E\)-convex set-valued mapping is presented.\(k\)UKK property in Banach spaces.https://zbmath.org/1449.460162021-01-08T12:24:00+00:00"Fan, Liying"https://zbmath.org/authors/?q=ai:fan.liying"Song, Jingjing"https://zbmath.org/authors/?q=ai:song.jingjing"Zhang, Jianing"https://zbmath.org/authors/?q=ai:zhang.jianingSummary: A new geometric property of Banach space \(k\)UKK is given. It is proved that Banach space with this property has weak Banach-Saks property, Banach space \(X\) is \(k\)NUC if and only if it is reflexive and has \(k\)UKK property. Considering the important role of geometric constants in Banach space geometric properties, the definition of the new constant \({R_2} (X) < k\) is given by the definition of \(k\)UKK and it is proved that when \({R_2} (X) < k\), the Banach space \(X\) has a weak fixed point property. Finally, the specific values are calculated in the Cesaro sequence space.Baire categorical aspects of first passage percolation. II.https://zbmath.org/1449.540422021-01-08T12:24:00+00:00"Maga, B."https://zbmath.org/authors/?q=ai:maga.balazsSummary: In this paper we continue our earlier work [\textit{B. Maga}, Acta Math. Hung. 156, No. 1, 145--171 (2018; Zbl 1424.54065)] about topological first passage percolation and answer certain questions asked in our previous paper. Notably, we prove that apart from trivialities, in the generic configuration there exists exactly one geodesic ray, in the sense that we consider two geodesic rays distinct if they only share a finite number of edges. Moreover, we show that in the generic configuration any not too small and not too large convex set arises as the limit of a sequence \(B(t_n)/t_n\) for some \(t_n\to\infty\). Finally, we define topological Hilbert first passage percolation, and amongst others we prove that certain geometric properties of the percolation in the generic configuration guarantee that we consider a setting linearly isomorphic to the ordinary topological first passage percolation.Boundedness of Schrödinger type operators on generalized Morrey spaces.https://zbmath.org/1449.351942021-01-08T12:24:00+00:00"Ren, Feng"https://zbmath.org/authors/?q=ai:ren.fengSummary: In this paper, we study Schrödinger type operators \({V^{\beta_1}}\nabla (-\Delta + V)^{-\beta_2}\). By the decomposition technique of function, the boundedness of these operators on generalized Morrey spaces is obtained when the potential \(V\) belongs to the reverse Hölder class \(RH_s\) with \({s > n/2}\). The results enrich and improve some existing ones.On the convergence of sequences of probability measures.https://zbmath.org/1449.280052021-01-08T12:24:00+00:00"Zaharopol, Radu"https://zbmath.org/authors/?q=ai:zaharopol.raduIn this paper \((X, d)\) is a Polish space, \(\mathcal{M}(X)\) all real valued Borel measures on \(X\) and \(C_b(X)\) all bounded continuous functions on \(X\). A function \(f\in C_b(X)\) is said to have bounded support if the set \(\overline{\{x\in X:f(x)\neq 0\}}\) is contained in an open ball. \(C^{(b)}_{bs}(X)\) are those functions in \(C_b(X)\) which have bounded support and \(C^{(ucb)}_{bs}(X)\) are those in \(C^{(b)}_{bs}(X)\) which are uniformly continuous. For a non-negative \(\nu\in\mathcal{M}(X)\), a Borel set \(A\subset X\) is said to be \(\nu\)-continuous if \(\nu(\bar A) =\nu(A^\circ)\). In this paper the author gives an extension of the Portmanteau Theorem. The main result is: Let \(\{\mu_n\}\) be a sequence of probability measures in \(\mathcal{M}(X)\) and \(\mu\in \mathcal{M}(X), \mu\ge 0\). Then following statements are equivalent: (a) \(\mu_n\to\mu\) pointwise on \(C^{(b)}_{bs}(X)\); (b) \(\mu_n\to\mu\) pointwise on \(C^{(ucb)}_{bs}(X)\); (c) for every closed bounded set \(F\subset X\) we have lim sup \(\mu_n(F) \le \mu(F)\), and for every open bounded set \(G\subset X\) we have lim inf \(\mu_n(G) \ge \mu(G)\); (d) \(\mu_n(A)\to\mu(A)\) for every \(\mu\)-continuous Borel set \(A\subset X\). Many sophisticated lemmas are proved to reach this result.
Reviewer: Surjit Singh Khurana (Iowa City)The Laplace' quasi-operator in quasi-Sobolev spaces.https://zbmath.org/1449.460262021-01-08T12:24:00+00:00"Al-Delfi, Jawad Kadim K."https://zbmath.org/authors/?q=ai:al-delfi.jawad-kadim-kSummary: The quasi-Sobolev spaces notion introduced in the article is based on the quasi-norm concept. Completeness of these spaces on the appropriate quasi-norms is proved and the continuous embedding of these spaces is shown in the work. Also Laplace' and Green's quasi-operators concepts are introduced; it is shown that these quasi-operators are toplinear isomorphisms.Riesz triple almost lacunary \(\chi^3\) sequence spaces defined by a Orlicz function. I.https://zbmath.org/1449.460092021-01-08T12:24:00+00:00"Subramanian, N."https://zbmath.org/authors/?q=ai:subramanian.nagarajan"Esi, A."https://zbmath.org/authors/?q=ai:esi.ayhan"Aiyub, M."https://zbmath.org/authors/?q=ai:aiyub.mohammadSummary: In this paper we introduce a new concept for Riesz almost lacunary \({\chi}^3\) sequence spaces strong \(P\)-convergent to zero with respect to an Orlicz function and examine some properties of the resulting sequence spaces. We introduce and study statistical convergence of Riesz almost lacunary \({\chi}^3\) sequence spaces and some inclusion theorems are discussed.Maximally unitarily mixed states on a \(C^*\)-algebra.https://zbmath.org/1449.460542021-01-08T12:24:00+00:00"Archbold, Robert"https://zbmath.org/authors/?q=ai:archbold.robert-j"Robert, Leonel"https://zbmath.org/authors/?q=ai:robert.leonel"Tikuisis, Aaron"https://zbmath.org/authors/?q=ai:tikuisis.aaron-peterSummary: We investigate the set of maximally mixed states of a \(C^*\)-algebra, extending previous work by Alberti on von Neumann algebras. We show that, unlike for von Neumann algebras, the set of maximally mixed states of a \(C^*\)-algebra may fail to be weak\(^*\) closed. We obtain, however, a concrete description of the weak\(^*\) closure of this set, in terms of tracial states and states which factor through simple traceless quotients. For \(C^*\)-algebras with the Dixmier property or with Hausdorff primitive spectrum we are able to advance our investigations further. We pose several questions.