Recent zbMATH articles in MSC 47Bhttps://zbmath.org/atom/cc/47B2021-05-28T16:06:00+00:00WerkzeugNorm-controlled inversion of Banach algebras of infinite matrices.https://zbmath.org/1459.460472021-05-28T16:06:00+00:00"Fang, Qiquan"https://zbmath.org/authors/?q=ai:fang.qiquan"Shin, Chang Eon"https://zbmath.org/authors/?q=ai:shin.chang-eonThe authors study the properties of the Baskakov-Gohberg-Sjöstrand algebras, which are certain Banach algebras of infinite matrices.
They prove some technical results about the inequalities describing the norm of the inverse of an element of this algebra.
Reviewer: Mart Abel (Tartu)Toeplitz operators with discontinuous symbols on the sphere.https://zbmath.org/1459.470122021-05-28T16:06:00+00:00"Barron, Tatyana"https://zbmath.org/authors/?q=ai:barron.tatyana"Itkin, David"https://zbmath.org/authors/?q=ai:itkin.davidMost known results on Berezin-Toeplitz operators \(T_f^{(k)}\) were obtained for \(C^{\infty}\) symbols \(f\). In the paper under review, some explicit calculations are presented for the Toeplitz operators with particular discontinuous symbols on the 2-dimensional torus.
For the entire collection see [Zbl 1364.22001].
Reviewer: Michal Zajac (Bratislava)The case a nonselfadjoint problem with a posteriori choice of parameter regularization for implicit iteration method for the solution of linear equations with approximate operator.https://zbmath.org/1459.650682021-05-28T16:06:00+00:00"Matysik, O. V."https://zbmath.org/authors/?q=ai:matysik.o-vSummary: The implicit iteration method for solution of the first-kind operator equations with a non self-adjoint bounded operator in Hilbert space is proposed. Convergence of a method is proved in case of an a posteriori choice of number of iterations in the usual norm of Hilbert space, supposing that not only the right part of the equation but the operator as well have errors. An estimation of the error of the method and of the a posteriori moment of stop are found.Inverse problems for Jacobi operators. IV: Interior mass-spring perturbations of semi-infinite systems.https://zbmath.org/1459.470142021-05-28T16:06:00+00:00"del Rio, Rafael"https://zbmath.org/authors/?q=ai:del-rio-castillo.rafael-rene"Kudryavtsev, Mikhail"https://zbmath.org/authors/?q=ai:kudryavtsev.mikhail"Silva, Luis O."https://zbmath.org/authors/?q=ai:silva.luis-oThe authors consider a linear semi-infinite mass-spring system which is modeled by a Jacobi operator \(J\) associated with the Jacobi matrix
\[
\begin{pmatrix} q_1 & b_1 & 0 & 0 & \dots \\ b_1 & q_2 & b_2 & 0 & \dots \\ 0 & b_2 & q_3 & b_3 & \dots \\ 0 & 0 & b_3 & q_4 & \dots\\ \vdots & \vdots & \vdots & \vdots & \ddots \end{pmatrix},
\]
where \(q_j = -\frac{k_{j+1} +k_j}{m_j}\) and \(b_j = \frac{k_{j+1}}{\sqrt{m_j m_{j+1}}}\) for \(j \in \mathbb{N}\). Here, \(\{m_j\}_{j=1}^{\infty}\) are the masses and \(\{k_j\}_{j=1}^{\infty}\) are the spring constants. The following inverse problem is solved: Given the spectrum of \(J\) as well as the spectrum of \(\tilde{J}_n\) (a perturbation of \(J\)), the matrix entries corresponding to \(J\) can be determined. In order to solve this inverse problem, the characterization of the relative distribution of the spectra of \(J\) and \(\tilde{J}_n\) is used. The Green functions of both the original and the perturbed operators play a fundamental role in the analysis. It should be noted that necessary and sufficient conditions for two sequences to be the spectra of the original mass-spring system and the perturbed mass-spring system are provided.
For Part I, see [\textit{R. del Rio} and \textit{M. Kudryavtsev}, Inverse Probl. 28, No. 5, Article ID 055007, 18 p. (2012; Zbl 1259.47040)], for Part II, see [\textit{R. del Rio} et al., J. Math. Phys. Anal. Geom. 9, No. 2, 165--190 (2013; Zbl 1305.47022)], and for Part III, see [\textit{R. del Rio} et al., Inverse Probl. Imaging 6, No. 4, 599--621 (2012; Zbl 1264.47032)].
Reviewer: Sonja Currie (Wits)Some 20+ year old problems about Banach spaces and operators on them.https://zbmath.org/1459.460102021-05-28T16:06:00+00:00"Johnson, William B."https://zbmath.org/authors/?q=ai:johnson.william-bWilliam B. Johnson has made made profound contributions to Banach space theory for 50+~years. In this survey, he presents solutions to long-standing problems he was involved in. The topics include:
-- the diameter of the isomorphism class of a separable Banach space;
-- commutators on Banach spaces;
-- the number of closed ideals in \(L(L_p)\) [here one should add the recent paper [\textit{W.~B. Johnson} and \textit{G.~Schechtman}, ``The number of closed ideals in \(L(L_p)\)'', Preprint, \url{arXiv:2003.11414}] showing that for \(1<p\neq2<\infty\) the number of closed ideals is \(2^{2^{\aleph_0}}\), which was still open at the time of writing the paper under review];
-- spaces that are uniformly homeomorphic to \(\mathscr{L}_1\)-spaces;
-- weakly null sequences in \(L_1\);
-- subspaces of spaces with an unconditional bases;
-- Tauberian operators on \(L_1\);
-- variants of the approximation property.
For proofs of the results, the readers are referred to the original articles cited in the paper.
For the entire collection see [Zbl 1437.00045].
Reviewer: Dirk Werner (Berlin)Upper bound of the third Hankel determinant for a subclass of \(q\)-starlike functions associated with the \(q\)-exponential function.https://zbmath.org/1459.050202021-05-28T16:06:00+00:00"Srivastava, H. M."https://zbmath.org/authors/?q=ai:srivastava.hari-mohan"Khan, Bilal"https://zbmath.org/authors/?q=ai:khan.bilal"Khan, Nazar"https://zbmath.org/authors/?q=ai:khan.nazar"Tahir, Muhammad"https://zbmath.org/authors/?q=ai:tahir.muhammad-ateeq|tahir.muhammad-hussain|tahir.muhammad-atif"Ahmad, Sarfraz"https://zbmath.org/authors/?q=ai:ahmad.sarfraz"Khan, Nasir"https://zbmath.org/authors/?q=ai:khan.nasir-saeed|khan.nasir-mSummary: By making use of the concept of basic (or \(q\)-) calculus, a subclass \(\mathcal{S}^\ast ( \mathcal{L} , q )\) of \(q\)-starlike functions, which is associated with the \(q\)-exponential function, is introduced here in the open unit disk \(\mathbb{U}\) given by \(\mathbb{U} = \{ z : z \in \mathbb{C} \text{ and } | z | < 1 \}\). The main object of this article is to determine the upper bound of the third-order Hankel determinant \(H_3(1)\) for functions belonging to the \(q\)-starlike function class \(\mathcal{S}^\ast ( \mathcal{L} , q )\). For validity of our results, relevant connections with those in earlier works are also pointed out.Aleksandrov-Clark theory for Drury-Arveson space.https://zbmath.org/1459.460362021-05-28T16:06:00+00:00"Jury, M. T."https://zbmath.org/authors/?q=ai:jury.michael-t"Martin, R. T. W."https://zbmath.org/authors/?q=ai:martin.robert-t-wSummary: Recent work has demonstrated that Clark's theory of unitary perturbations of the backward shift on a deBranges-Rovnyak space on the disk has a natural extension to the several-variable setting. In the several-variable case, the appropriate generalization of the Schur class of contractive analytic functions is the closed unit ball of the Drury-Arveson multiplier algebra and the Aleksandrov-Clark measures are necessarily promoted to positive linear functionals on a symmetrized subsystem of the free disk operator system \(\mathcal {A} _d + \mathcal {A} _d ^*\), where \(\mathcal {A} _d\) is the free or non-commutative disk algebra on \(d\) generators. We continue this program for vector-valued Drury-Arveson space by establishing the existence of a canonical `tight' extension of any Aleksandrov-Clark map to the full free disk operator system. We apply this tight extension to generalize several earlier results and we characterize all extensions of the Aleksandrov-Clark maps.Copies of \(l_p^n\)'s uniformly in the spaces \(\Pi_2(C[0,1],X)\) and \(\Pi_1(C[0,1],X)\).https://zbmath.org/1459.460142021-05-28T16:06:00+00:00"Popa, Dumitru"https://zbmath.org/authors/?q=ai:popa.dumitruSummary: We study the presence of copies of \(l_p^n\)'s uniformly in the spaces \(\Pi_2(C[0,1],X)\) and \(\Pi_1(C[0,1],X)\). By using Dvoretzky's theorem we deduce that if \(X\) is an infinite-dimensional Banach space, then \(\Pi_2(C[0,1],X)\) contains \(\lambda\sqrt{2}\)-uniformly copies of \(l_{\infty}^n\)'s and \(\Pi_1(C[0,1],X)\) contains \(\lambda\)-uniformly copies of \(l_2^n\)'s for all \(\lambda >1\). As an application, we show that if \(X\) is an infinite-dimensional Banach space then the spaces \(\Pi_2(C[0,1],X)\) and \(\Pi_1(C[0,1],X)\) are distinct, extending the well-known result that the spaces \(\Pi_2(C[0,1],X)\) and \(\mathcal{N}(C[0,1],X)\) are distinct.Continuity and compactness for pseudo-differential operators with symbols in quasi-Banach spaces or Hörmander classes.https://zbmath.org/1459.470192021-05-28T16:06:00+00:00"Toft, Joachim"https://zbmath.org/authors/?q=ai:toft.joachimThere are several Schatten-von Neumann results for pseudo-differential operators with symbols in modulation spaces, Besov spaces, and Sobolev spaces. In this paper, the author deduces continuity and Schatten-von Neumann properties for operators with matrices satisfying mixed quasi-norm estimates with Lebesgue and Schatten parameters in \((0,\infty)\). These results are used to deduce continuity and Schatten-von Neumann properties for pseudo-differential operators with symbols in quasi-Banach modulation spaces or in appropriate Hörmander classes. As an application, the author gives some examples and other results for Schatten-von Neumann symbols.
Reviewer: Ahmed Lesfari (El Jadida)On the law of large numbers for compositions of independent random semigroups.https://zbmath.org/1459.470162021-05-28T16:06:00+00:00"Sakbaev, V. Zh."https://zbmath.org/authors/?q=ai:sakbaev.vsevolod-zhSummary: We study random linear operators in Banach spaces and random one-parameter semigroups of such operators. For compositions of independent random semigroups of linear operators in the Hilbert space, we obtain sufficient conditions for fulfilment of the law of large numbers and give examples of its violation.On a Riesz basis of exponentials related to a family of analytic operators and application.https://zbmath.org/1459.460162021-05-28T16:06:00+00:00"Ellouz, Hanen"https://zbmath.org/authors/?q=ai:ellouz.hanen"Feki, Ines"https://zbmath.org/authors/?q=ai:feki.ines"Jeribi, Aref"https://zbmath.org/authors/?q=ai:jeribi.arefSummary: In this paper, we are interested in the perturbed operator
\[ T(\varepsilon) := T_0+\varepsilon T_1 +\varepsilon^2T_2+\ldots +\varepsilon^k T_k+\cdots \]
where \(\varepsilon \in \mathbb{C}\), \(T_0\) is a closed densely defined linear operator on a separable Hilbert space \(\mathcal{H}\) with domain \(\mathcal{D}(T_0)\) having isolated eigenvalues with multiplicity one ,whereas \(T_1, T_2,\ldots\) are linear operators on \(\mathcal{H}\) having the same domain \(\mathcal{D}\supset \mathcal{D}(T_0)\) and satisfying a specific growing inequality. The basic idea here is to investigate under sufficient conditions the existence of Riesz bases of exponentials, where the exponents corresponding as a sequence of eigenvalues of \(T(\varepsilon )\) can be developed as entire series of \(\varepsilon \).
An application to a nonself-adjoint problem describing the radiation of a vibrating structure in a light fluid is presented.Quantitative compactness estimate for scalar conservation laws with non-convex fluxes.https://zbmath.org/1459.352822021-05-28T16:06:00+00:00"Ancona, Fabio"https://zbmath.org/authors/?q=ai:ancona.fabio"Glass, Olivier"https://zbmath.org/authors/?q=ai:glass.olivier"Nguyen, Khai T."https://zbmath.org/authors/?q=ai:nguyen.khai-tSummary: This note provides a survey of recent results establishing upper and lower estimates for the Kolmogorov \(\varepsilon\)-entropy of the image through the mapping \(S_t\) of bounded sets in \(L^1\cap L^\infty\) for scalar conservation laws with non-convex fluxes in one space dimension. As suggested by \textit{P. D. Lax} [Recent advances in numerical analysis, Proc. Symp., Madison/Wis. 1978, 107--117 (1978; Zbl 0457.65068)], these quantitative compactness estimates could provide a measure of the order of ``resolution'' of the numerical methods implemented for these equations.
For the entire collection see [Zbl 1453.35003].Supersymmetric Fibonacci polynomials.https://zbmath.org/1459.810482021-05-28T16:06:00+00:00"Yamani, Hashim A."https://zbmath.org/authors/?q=ai:yamani.hashim-aSummary: It has long been recognized that Fibonacci-type recurrence relations can be used to define a set of versatile polynomials \(\{p_n (z)\}\) that have Fibonacci numbers and Chebyshev polynomials as special cases. We show that a tridiagonal matrix, which can be factored into the product \(AB\) of two special matrices \(A\) and \(B\), is associated with these polynomials. We apply tools that have been developed to study the supersymmetry of Hamiltonians that have a tridiagonal matrix representation in a basis to derive a set of partner polynomials \(\{ p_n^{(+)} (z)\}\) associated with the matrix product \(BA\). We find that special cases of these polynomials share similar properties with the Fibonacci numbers and Chebyshev polynomials. As a result, we find two new sum rules that involve the Fibonacci numbers and their product with Chebyshev polynomials.On the Bonsall cone spectral radius and the approximate point spectrum.https://zbmath.org/1459.470212021-05-28T16:06:00+00:00"Müller, Vladimir"https://zbmath.org/authors/?q=ai:muller.vladimir"Peperko, Aljoša"https://zbmath.org/authors/?q=ai:peperko.aljosaAuthor's abstract: We study the Bonsall cone spectral radius and the approximate point spectrum of (in general nonlinear) positively homogeneous, bounded and supremum preserving maps, defined on a max-cone in a given normed vector lattice. We prove that the Bonsall cone spectral radius of such maps is always included in its approximate point spectrum. Moreover, the approximate point spectrum always contains a (possibly trivial) interval. Our results apply to a large class of (nonlinear) max-type operators. We also generalize a~known result that the spectral radius of a positive (linear) operator on a~Banach lattice is contained in the approximate point spectrum. Under additional generalized compactness type assumptions, our results imply Krein-Rutman type results.
Reviewer: Mohamed Ali Toumi (Bizerte)Spaces generated by the cone of sublinear operators.https://zbmath.org/1459.460042021-05-28T16:06:00+00:00"Slimane, A."https://zbmath.org/authors/?q=ai:slimane.aSummary: This paper deals with a study on classes of non linear operators. Let \(SL(X,Y)\) be the set of all sublinear operators between two Riesz spaces \(X\) and \(Y\). It is a convex cone of the space \(H(X,Y)\) of all positively homogeneous operators. In this paper we study some spaces generated by this cone, therefore we study several properties, which are well known in the theory of Riesz spaces, like order continuity, order boundedness etc. Finally, we try to generalise the concept of adjoint operator. First, by using the analytic form of Hahn-Banach theorem, we adapt the notion of adjoint operator to the category of positively homogeneous operators. Then we apply it to the class of operators generated by the sublinear operators.The spectral spread of Hermitian matrices.https://zbmath.org/1459.420442021-05-28T16:06:00+00:00"Massey, Pedro"https://zbmath.org/authors/?q=ai:massey.pedro-g"Stojanoff, Demetrio"https://zbmath.org/authors/?q=ai:stojanoff.demetrio"Zárate, Sebastián"https://zbmath.org/authors/?q=ai:zarate.sebastianSummary: Let \(A\) be an \(n\times n\) complex Hermitian matrix and let \(\lambda(A)=(\lambda_1,\dots,\lambda_n)\in\mathbb{R}^n\) denote the eigenvalues of \(A\), counting multiplicities and arranged in non-increasing order. Motivated by problems arising in the theory of low rank matrix approximation, we study the spectral spread of \(A\), denoted \(\text{Spr}^+(A)\), given by \(\text{Spr}^+(A)=(\lambda_1-\lambda_n, \lambda_2-\lambda_{n-1},\dots,\lambda_k-\lambda_{n-k+1})\in \mathbb{R}^k\), where \(k=[n/2]\) (integer part). The spectral spread is a vector-valued measure of dispersion of the spectrum of \(A\), that allows one to obtain several submajorization inequalities. In the present work we obtain inequalities that are related to Tao's inequality for anti-diagonal blocks of positive semidefinite matrices, Zhan's inequalities for the singular values of differences of positive semidefinite matrices, extremal properties of direct rotations between subspaces, generalized commutators and distances between matrices in the unitary orbit of a Hermitian matrix.Compactness of Hankel operators with symbols continuous on the closure of pseudoconvex domains.https://zbmath.org/1459.470132021-05-28T16:06:00+00:00"Clos, Timothy G."https://zbmath.org/authors/?q=ai:clos.timothy-g"Çelik, Mehmet"https://zbmath.org/authors/?q=ai:celik.mehmet"Şahutoğlu, Sönmez"https://zbmath.org/authors/?q=ai:sahutoglu.sonmezThe authors study the relation between the compactness of a Hankel operator and the boundary behavior of the symbol. Let \(\Omega\) be a domain in \(\mathbb{C}^n\) and let \(A^2(\Omega)\) be its Bergman space. The Bergman projection is the orthogonal projection \(P:L^2(\Omega)\to A^2(\Omega)\). The Hankel operator \(H_{\phi}:A^2(\Omega)\to L^2(\Omega)\) with symbol \(\phi\in L^{\infty}(\Omega)\) is defined as
\[
H_{\phi}f=(I-P)(\phi f),
\]
where \(I\) denotes the identity operator.
The main result is that, when \(\Omega\) is bounded pseudoconvex in \(\mathbb{C}^2\) with Lipschitz boundary and \(\phi\in C(\overline{\Omega})\), if \(H_{\phi}\) is compact on \(A^2(\Omega)\), then \(\phi\circ f\) is holomorphic for any holomorphic \(f:\mathbb{D}\to b\Omega\). This improves the previous result by \textit{T. G. Clos} and \textit{S. Şahutoğlu} [Complex Anal. Oper. Theory 12, No. 2, 365--376 (2018; Zbl 06837878)] in one direction. Based on the result by \textit{P. Matheos} [A Hartogs domain with no analytic discs in the boundary for which the $\overline{\partial}$-Neumann problem is not compact. PhD Thesis. University of California Los Angeles, CA (1997)] and the result by \textit{S. Şahutoğlu} and \textit{Y. E. Zeytuncu} [J. Geom. Anal. 27, No. 2, 1274--1285 (2017; Zbl 1375.32061)], the converse the main result is not true.
The authors also prove a similar result in the setting of convex domains in \(\mathbb{C}^n\). Namely, when \(\Omega\) is bounded convex in \(\mathbb{C}^n\) and \(\phi\in C(\overline{\Omega})\), if \(H_{\phi}\) is compact on \(A^2(\Omega)\), then \(\phi\circ f\) is holomorphic for any holomorphic \(f:\mathbb{D}\to b\Omega\). However, the converse of the second result is still open.
Reviewer: Liwei Chen (Columbus)Hausdorff moment sequences induced by rational functions.https://zbmath.org/1459.440052021-05-28T16:06:00+00:00"Reza, Md. Ramiz"https://zbmath.org/authors/?q=ai:reza.md-ramiz"Zhang, Genkai"https://zbmath.org/authors/?q=ai:zhang.genkaiSummary: We study the Hausdorff moment problem for a class of sequences, namely \((r(n))_{n\in\mathbb{Z}_+}\), where \(r\) is a rational function in the complex plane. We obtain a necessary condition for such sequence to be a Hausdorff moment sequence. We found an interesting connection between Hausdorff moment problem for this class of sequences with finite divided differences and convolution of complex exponential functions. We provide a sufficient condition on the zeros and poles of a rational function \(r\) so that \((r(n))_{n\in\mathbb{Z}_+}\) is a Hausdorff moment sequence. G. Misra asked whether the module tensor product of a subnormal module with the Hardy module over the polynomial ring is again a subnormal module or not. Using our necessary condition we answer the question of G. Misra in negative. Finally, we obtain a characterization of all real polynomials \(p\) of degree up to 4 and a certain class of real polynomials of degree 5 for which the sequence \((1/p(n))_{n\in\mathbb{Z}_+}\) is a Hausdorff moment sequence.Banach-space operators acting on semicircular elements induced by orthogonal projections.https://zbmath.org/1459.460532021-05-28T16:06:00+00:00"Cho, Ilwoo"https://zbmath.org/authors/?q=ai:cho.ilwooSummary: The main purposes of this paper are (i) to construct-and-study weighted-semicircular elements from mutually orthogonal \(|\mathbb{Z}|\)-many projections, and the Banach \(*\)-probability space \(\mathbb{L}_Q\) generated by these operators, (ii) to establish \(*\)-isomorphisms on \(\mathbb{L}_Q\) induced by shifting processes on the set \(\mathbb{Z}\) of integers, (iii) to consider how the \(*\)-isomorphisms of (ii) generate Banach-space adjointable operators acting on the Banach \(*\)-algebra \(\mathbb{L}_Q\), (iv) to investigate operator-theoretic properties of the operators of (iii), and (v) to study how the Banach-space operators of (iii) distort the original free-distributional data on \(\mathbb{L}_Q\). As application, one can check how the semicircular law is distorted by our Banach-space operators on \(\mathbb{L}_Q\).Extension properties of some completely positive maps.https://zbmath.org/1459.470152021-05-28T16:06:00+00:00"Sun, Xiuhong"https://zbmath.org/authors/?q=ai:sun.xiuhong"Li, Yuan"https://zbmath.org/authors/?q=ai:li.yuan.2Suppose that \(B( \mathcal H)\), \(K( \mathcal H)\) and \(T( \mathcal H)\) are the sets of all bounded linear operators, compact operators and trace-class operators on the Hilbert space \({\mathcal H}\), respectively. A~completely positive map is a map \(\Phi: B(\mathcal H) \to B(\mathcal K)\) between \(C^*\)-algebras with the property that, for each~\(n\), \(\Phi\) is \(n\)-positive in the sense that the map \(\Phi_n: \mathbb{M}_n(B(\mathcal H)) \to \mathbb{M}_n(B(\mathcal K))\) defined by \(\Phi([a_{ij}])=[\Phi(a_{ij})]\) is positive, where \(\mathbb{M}_n(B(\mathcal H))\) and \(\mathbb{M}_n(B(\mathcal K))\) are the \(C^*\)-algebras of all \(n\times n\) matrices with entries in \(B(\mathcal H)\) and \(B(\mathcal K)\), respectively. A~normal completely positive map is a contraction map \(\Phi: B(\mathcal H) \to B(\mathcal K)\) if there exists a sequence \(\{A_i \}\) of \(B({\mathcal H},{\mathcal K})\) such that \( \Phi(X)= \Sigma_i A_i X A_{i}^{*}\), where \(\Sigma_i A_i A_{i}^{*} \leq I\) and \(X \in B({\mathcal H},{\mathcal K})\). In finite-dimensional Hilbert spaces \( \mathcal H\) and \( \mathcal K\), equivalent conditions for completely positive maps were obtained by \textit{M.-D. Choi} [Linear Algebra Appl. 10, 285--290 (1975; Zbl 0327.15018)].
In the present paper, the authors consider expressions and extension properties of completely positive maps from \(K( {\mathcal H})\) to \( K( {\mathcal K})\) and from \(T( {\mathcal H})\) to \(K( {\mathcal K})\). They study the relationship between normal completely positive maps and completely positive maps, too.
Reviewer: Kamran Sharifi (Shahrood)Schur operators and domination problem.https://zbmath.org/1459.470082021-05-28T16:06:00+00:00"Baklouti, Hamadi"https://zbmath.org/authors/?q=ai:baklouti.hamadi"Hajji, Mohamed"https://zbmath.org/authors/?q=ai:hajji.mohamed-ali|hajji.mohamed-karimLet \(X,Y\) be a Banach spaces. The authors call \(T \in {\mathcal L}(X,Y)\) a Schur operator if, for any closed subspace \(M \subset X\), \(T|M\) has a bounded inverse implies that \(M\) contains an infinite-dimensional closed subspace with the Schur property. The authors show that these operators form a closed two-sided ideal in \({\mathcal L}(X)\). If, in addition, \(X,Y\) are Banach lattices, the authors investigate the question: when is being Schur preserved by domination? The authors show that, for positive operators \(R_i \leq T_i: E_i \to E_{i+1}\) for \(1 \leq i \leq 4\), if \(T_1,T_3\) are Schur and \(T_2,T_4\) are order weakly compact, then \(R_4R_3R_2R_1\) is Schur.
Reviewer: T.S.S.R.K. Rao (Bangalore)Correction to: ``Surjectivity of Hadamard type operators on spaces of smooth functions''.https://zbmath.org/1459.460312021-05-28T16:06:00+00:00"Domański, Paweł"https://zbmath.org/authors/?q=ai:domanski.pawel"Langenbruch, Michael"https://zbmath.org/authors/?q=ai:langenbruch.michaelCorrection to the authors' paper [ibid. 113, No. 2, 1625--1676 (2019; Zbl 1437.46030)].Construct approximate dual g-frames in Hilbert spaces.https://zbmath.org/1459.420432021-05-28T16:06:00+00:00"Guo, Qianping"https://zbmath.org/authors/?q=ai:guo.qianping"Leng, Jingsong"https://zbmath.org/authors/?q=ai:leng.jingsong"Li, Houbiao"https://zbmath.org/authors/?q=ai:li.houbiaoSummary: In this paper, we first present some simple approaches to obtain dual and approximate dual g-frames. Then, we show that approximate dual g-frames are stable under some conditions. Finally, we give mainly a new characterization for approximate dual g-frames associated with given g-frames and bounded operators. Moreover, we prove that if two g-frames are close to each other, then we can find approximate dual g-frames associated with them which are close to each other.The inverse monodromy problem.https://zbmath.org/1459.460352021-05-28T16:06:00+00:00"Arov, Damir Z."https://zbmath.org/authors/?q=ai:arov.damir-zyamovich"Dym, Harry"https://zbmath.org/authors/?q=ai:dym.harrySummary: The inverse monodromy problem for \(m \times m\) canonical differential systems \( y'_{t}(\lambda) = i\lambda y_{t} (\lambda)H(t)J \) on a finite interval \([0, d]\) is to recover the Hamiltonian \(H(t)\) of the differential system from the monodromy matrix, i.e., the value of the matrizant (fundamental solution) of the system at the right-hand end point \(d\) of the interval. This problem does not have a unique solution unless extra constraints are imposed. A number of known results are reviewed briefly. Special classes of monodromy matrices for which the solutions of the inverse monodromy problem may be parametrized by identifying the matrizant with the resolvent matrices of a class of bitangential extension problems are discussed. The exposition makes extensive use of two classes of reproducing kernel Hilbert spaces of vector-valued entire functions that originate in the work of Louis de Branges and the interplay between them. Some new subclasses of these spaces are introduced and
their role in the inverse monodromy problem are discussed.
For the entire collection see [Zbl 1388.46001].Multilinear estimates for Calderón commutators.https://zbmath.org/1459.420202021-05-28T16:06:00+00:00"Lai, Xudong"https://zbmath.org/authors/?q=ai:lai.xudongSummary: In this paper, we investigate the multilinear boundedness properties of the higher (\(n\)-th) order Calderón commutator for dimensions larger than two. We establish all multilinear endpoint estimates for the target space \(L^{\frac{d}{d+n},\infty }(\mathbb{R}^d)\), including that Calderón commutator maps the product of Lorentz spaces \(L^{d,1}(\mathbb{R}^d)\times \cdots \times L^{d,1}(\mathbb{R}^d)\times L^1(\mathbb{R}^d)\) to \(L^{\frac{d}{d+n},\infty }(\mathbb{R}^d)\), which is the higher dimensional nontrivial generalization of the endpoint estimate that the \(n\)-th order Calderón commutator maps \(L^1(\mathbb{R})\times \cdots \times L^1(\mathbb{R})\times L^1(\mathbb{R})\) to \(L^{\frac{1}{1+n},\infty }(\mathbb{R})\). When considering the target space \(L^r(\mathbb{R}^d)\) with \(r<\frac{d}{d+n} \), some counterexamples are given to show that these multilinear estimates may not hold. The method in the present paper seems to have a wide range of applications and it can be applied to establish the similar results for Calderón commutator with a rough homogeneous kernel.A note on inner-outer factorization of wide matrix-valued functions.https://zbmath.org/1459.470112021-05-28T16:06:00+00:00"Frazho, A. E."https://zbmath.org/authors/?q=ai:frazho.arthur-e"Ran, A. C. M."https://zbmath.org/authors/?q=ai:ran.andre-c-mSummary: In this paper, we expand some of the results of [\textit{A. E. Frazho} et al., Integral Equations Oper. Theory 66, No. 2, 215--229 (2010; Zbl 1210.47045); ibid 70, No. 3, 395--418 (2011; Zbl 1293.47027); Oper. Matrices 6, No. 4, 833--857 (2012; Zbl 1257.47032)]. In fact, using the same techniques, we provide formulas for the full rank inner-outer factorization of a wide matrix-valued rational function \(G\) with \(H^\infty\) entries, that is, functions \(G\) with more columns than rows. State space formulas are derived for the inner and outer factor of \(G\).
For the entire collection see [Zbl 1411.47002].Criterion for the functional dissipativity of second order differential operators with complex coefficients.https://zbmath.org/1459.351172021-05-28T16:06:00+00:00"Cialdea, A."https://zbmath.org/authors/?q=ai:cialdea.alberto"Maz'ya, V."https://zbmath.org/authors/?q=ai:mazya.vladimir-gilelevichSummary: In the present paper we consider the Dirichlet problem for the second order differential operator \(E = \nabla(\mathcal{A} \nabla)\), where \(\mathcal{A}\) is a matrix with complex valued \(L^\infty\) entries. We introduce the concept of dissipativity of \(E\) with respect to a given function \(\varphi : \mathbb{R}^+ \to \mathbb{R}^+\). Under the assumption that the \( \operatorname{\mathbb{I}m} \mathcal{A}\) is symmetric, we prove that the condition \(| s \varphi^\prime (s) | | \langle \operatorname{\mathbb{I}m} \mathcal{A} (x) \xi, \xi \rangle | \leqslant 2 \sqrt{ \varphi (s) [ s \varphi (s)]^\prime} \langle \operatorname{\mathbb{R}e} \mathcal{A} (x) \xi, \xi \rangle \) (for almost every \(x \in \Omega \subset \mathbb{R}^N\) and for any \(s > 0\), \(\xi \in \mathbb{R}^N\)) is necessary and sufficient for the functional dissipativity of \(E\).Parabolic invariant tori in quasi-periodically forced skew-product maps.https://zbmath.org/1459.370512021-05-28T16:06:00+00:00"Guan, Xinyu"https://zbmath.org/authors/?q=ai:guan.xinyu"Si, Jianguo"https://zbmath.org/authors/?q=ai:si.jianguo"Si, Wen"https://zbmath.org/authors/?q=ai:si.wenSummary: We consider the existence of parabolic invariant tori for a class of quasi-periodically forced analytic skew-product maps \(\varphi : \mathbb{R}^n \times \mathbb{T}^d \to \mathbb{R}^n \times \mathbb{T}^d\):
\[
\varphi \begin{pmatrix} z \\ \theta \end{pmatrix} = \begin{pmatrix} z + \phi (z) + h (z, \theta) + \epsilon f (z, \theta) \\ \theta + \omega \end{pmatrix},
\]
where \(\phi : \mathbb{R}^n \to \mathbb{R}^n\) is a homogeneous function of degree \(l\) with \(l \geq 2\) and \(h = \mathcal{O}(|z|^{l + 1})\). We obtain the following results: (a) For \(n = 1, l\) being odd and \(\varepsilon\) sufficiently small, parabolic invariant tori exist if \(\omega\) satisfies the Brjuno-Rüssmann's non-resonant condition. (b) For \(n = 1\), and \(\varepsilon\) sufficiently small, parabolic invariant tori also exist if one of the following conditions holds: (i) First order average is non-zero, first order non-average part is small enough and the forcing frequency \(\omega\) does not need any arithmetic condition. (ii) First order average is non-zero and \(\omega\) satisfies the Brjuno-type weak non-resonant condition; (iii) \(l = 2\), first order average is zero, both first and second order non-average parts are small enough and \(\omega\) satisfying Brjuno-type weak non-resonant condition; (iv) \(l > 2\), first order average is zero, the second order average is non-zero, both first and second order non-average parts are small enough and \(\omega\) satisfies the Brjuno-type weak non-resonant condition. (c) In the case \(n > 1\), if first order average belongs to the interior of the range of \(\varphi, Spec(D \phi) \cap \text{i} \mathbb{R} = \emptyset\), first order non-average part is small enough and the forcing frequency \(\omega\) does not need any arithmetic condition, then the quasi-periodically forced skew-product maps above admit parabolic invariant tori for \(\varepsilon\) sufficiently small. The main methods of this paper are KAM theory and fixed point theorem, which are finally shown that it can be directly applied to the existence problem of quasi-periodic response solutions of degenerate harmonic oscillators.Fractional nonlinear dynamics of learning with memory.https://zbmath.org/1459.340462021-05-28T16:06:00+00:00"Tarasov, Vasily E."https://zbmath.org/authors/?q=ai:tarasov.vasily-eSummary: In this paper, we consider generalization of the Lucas model of learning (learning-by-doing) that is described in the paper [\textit{R. E. Lucas jun.}, Econometrica 61, No. 2, 251--272 (1993; Zbl 0825.90148)], who was awarded the Nobel Prize in Economic Sciences in 1995. The model equation is nonlinear differential equation of the first order used in macroeconomics to explain effects of innovation and technical change. In the standard learning model, the memory effects and memory fading are not taken into account. We propose the learning models that take into account fading memory. Fractional differential equations of the suggested models contain fractional derivatives with the generalized Mittag-Leffler function (the Prabhakar function) in the kernel and their special case containing the Caputo fractional derivative. These nonlinear fractional differential equations, which describe the learning-by-doing with memory, and the expressions of its exact solutions are suggested. Based on the exact solution of the model equation, we show that the estimated productivity growth rate can be changed by memory effects.Function theory and \(\ell^p\) spaces.https://zbmath.org/1459.460012021-05-28T16:06:00+00:00"Cheng, Raymond"https://zbmath.org/authors/?q=ai:cheng.raymond"Mashreghi, Javad"https://zbmath.org/authors/?q=ai:mashreghi.javad"Ross, William T."https://zbmath.org/authors/?q=ai:ross.william-t-junThis monograph is an up-to-date and well written compendium of results on \(\ell^p\)-spaces (\(p>0\)) as function spaces. The authors have contributed to various aspects of this theory over two decades. Their earlier monographs and long articles expounded both analytic and operator theoretic aspects. This Lecture Series will serve both beginning researchers and experts as one comprehensive, single source.
Starting with the discrete case and geometric aspects such as weak parallelogram laws, the book quickly gets to the main focus of studying \(\ell^p_A\), the space of analytic functions on the disk whose Taylor coefficients are in \(\ell^p\) for \( 0 < p \leq \infty\).
From Chapter~7 onwards, various standard operators on these spaces, such as the shift operator (invariant subspaces), isometries, composition operators, backward shift and multipliers are discussed.
The monograph has a very exhaustive list of references, covering the development of these ideas over a century.
Reviewer: T.S.S.R.K. Rao (Bangalore)Oscillation theory for the density of states of high dimensional random operators.https://zbmath.org/1459.810392021-05-28T16:06:00+00:00"Großmann, Julian"https://zbmath.org/authors/?q=ai:grossmann.julian-p"Schulz-Baldes, Hermann"https://zbmath.org/authors/?q=ai:schulz-baldes.hermann"Villegas-Blas, Carlos"https://zbmath.org/authors/?q=ai:villegas-blas.carlosSummary: Sturm-Liouville oscillation theory is studied for Jacobi operators with block entries given by covariant operators on an infinite dimensional Hilbert space. It is shown that the integrated density of states of the Jacobi operator is approximated by the winding of the Prüfer phase w.r.t. the trace per unit volume. This rotation number can be interpreted as a spectral flow in a von Neumann algebra with finite trace.On functional equations related to generalized Jordan derivations in rings.https://zbmath.org/1459.160392021-05-28T16:06:00+00:00"Nakkao, Sitthikorn"https://zbmath.org/authors/?q=ai:nakkao.sitthikorn"Leerawat, Utsanee"https://zbmath.org/authors/?q=ai:leerawat.utsaneeSummary: In this paper, we generalize the notions of Jordan derivation and generalized Jordan derivation to Jordan \((f, g)\)-derivation and generalized Jordan \((f, g)\)-derivation, respectively. Moreover, we investigate additive mappings satisfying some functional equations becomes Jordan \((f, g)\)-derivation and generalized Jordan \((f, g)\)-derivation.Dedekind complete unital $f$-algebras on which every band preserving operator is order bounded.https://zbmath.org/1459.060132021-05-28T16:06:00+00:00"Toumi, Mohamed Ali"https://zbmath.org/authors/?q=ai:toumi.mohamed-aliSummary: We give a complete description of those Dedekind complete unital $f$-algebras $A$ with the property that every band preserving operator on $A$ is order bounded. As an application, we furnish a new characterization of locally one (dimensional universally complete vector lattices.