Recent zbMATH articles in MSC 47Ahttps://www.zbmath.org/atom/cc/47A2022-01-14T13:23:02.489162ZWerkzeugFinite dimensional applications of the Dunford-Taylor integralhttps://www.zbmath.org/1475.150052022-01-14T13:23:02.489162Z"Caratelli, Diego"https://www.zbmath.org/authors/?q=ai:caratelli.diego"Palini, Ernesto"https://www.zbmath.org/authors/?q=ai:palini.ernesto"Ricci, Paolo Emilio"https://www.zbmath.org/authors/?q=ai:ricci.paolo-emilioSummary: The Dunford-Taylor integral is used in order to compute the inverse of a non-singular complex matrix. Then the obtained result is applied to derive the solution of some basic analytic problems as the solution of linear algebraic equations, the solution of matrix equations and of initial value problems for a linear system of ordinary differential equations.Eigenvalue paths arising from matrix pathshttps://www.zbmath.org/1475.150092022-01-14T13:23:02.489162Z"Jankowski, Eric"https://www.zbmath.org/authors/?q=ai:jankowski.eric"Johnson, Charles R."https://www.zbmath.org/authors/?q=ai:johnson.charles-richard-jun|johnson.charles-royalThe continuous variations in the entries of a complex square matrix induce continuous variations in its eigenvalues. If such a variation arises from a real parameter \(\alpha\in [0,1]\), then the eigenvalues follow continuous paths in the complex plane as \(\alpha\) shifts from \(0\) to \(1\).
In this paper, the authors study the nature of these eigenpaths, including their behavior under small perturbations of the matrix variations, as well as the resulting eigenpairings of the matrices that occur at \(\alpha=0\) and \(\alpha=1\).
It is proved that for any \(\varepsilon>0\) and for a matrix path \(C\), there is a \(\delta>0\) such that for any matrix path \(C^{\prime}\) with the norm difference of two paths less than \(\delta\) for any \(\alpha\in [0,1]\) and any \(C^{\prime}\) eigenpath set\(\{r^{\prime}_i\}^n\), there exists a \(C\) eigenpath set \(\{r_i\}^n\) satisfying \(|r_i(\alpha)-r^{\prime}_i(\alpha)|<\varepsilon.\) The authors also give analogs of their results in the setting of monic polynomials.
Reviewer: Kui Ji (Shijiazhuang)Some matrix inequalities related to Kwong functionshttps://www.zbmath.org/1475.150232022-01-14T13:23:02.489162Z"Zhao, Jianguo"https://www.zbmath.org/authors/?q=ai:zhao.jianguo"Wu, Junliang"https://www.zbmath.org/authors/?q=ai:wu.junliangA {Kwong function} is a function \(f:(a,b)\to\mathbb R\), with \(a\geq0\), such that the \(n\times n\) matrix
\[
\left[\frac{f(\lambda_k)+f(\lambda_j)}{\lambda_k+\lambda_j}\right]
\]
is positive semidefinite for any choice of distinct \(\lambda_1,\ldots,\lambda_n\in(a,b)\).
The authors use Kwong functions to generalize several well-known matrix norm inequalities. As an example, they show that if \(A,B,X\in M_n(\mathbb C)\) with \(A,B\) positive definite, and \(f,g:(0,\infty)\to\mathbb R\) are such that \(f/g\) is a Kwong function, then for any unitarily invariant norm \(\|\cdot\|_u\)
\[
\tfrac2{c}\,\|A^{1/2}XB^{1/2}\}_u\leq\|f(A)Xg(B)+g(A)Xf(B)\|_u
\]
holds, where \[c=\max\left\{\frac\lambda{f(\lambda)g(\lambda)}:\ \lambda\in\sigma(A)\cup\sigma(B)\right\}.\] In the case where \(f(x)=x^\mu\) and \(g(x)=x^{1-\mu}\), this particular case was proven by \textit{R. Bhatia} and \textit{C. Davis} [SIAM J. Matrix Anal. Appl. 14, No. 1, 132--136 (1993; Zbl 0767.15012)].
Reviewer: Martín Argerami (Regina)Nonparallel flat portions on the boundaries of numerical ranges of 4-by-4 nilpotent matriceshttps://www.zbmath.org/1475.150252022-01-14T13:23:02.489162Z"Cox, Mackenzie"https://www.zbmath.org/authors/?q=ai:cox.mackenzie"Grewe, Weston"https://www.zbmath.org/authors/?q=ai:grewe.weston"Hochrein, Grace"https://www.zbmath.org/authors/?q=ai:hochrein.grace"Patton, Linda"https://www.zbmath.org/authors/?q=ai:patton.linda-j"Spitkovsky, Ilya"https://www.zbmath.org/authors/?q=ai:spitkovsky.ilya-matveySummary: The 4-by-4 nilpotent matrices whose numerical ranges have nonparallel flat portions on their boundary that are on lines equidistant from the origin are characterized. Their numerical ranges are always symmetric about a line through the origin and all possible angles between the lines containing the flat portions are attained.Norm inequalities for positive semi-definite matrices and a question of Bourin. II.https://www.zbmath.org/1475.150262022-01-14T13:23:02.489162Z"Hayajneh, Mostafa"https://www.zbmath.org/authors/?q=ai:hayajneh.mostafa"Hayajneh, Saja"https://www.zbmath.org/authors/?q=ai:hayajneh.saja"Kittaneh, Fuad"https://www.zbmath.org/authors/?q=ai:kittaneh.fuadThroughout the paper, the authors consider all matrices to be \(n\times n\) complex matrices and denote by \(|\!|\!|\cdot |\!|\!|\) any unitarily invariant norm on the space of \(n\times n\) complex matrices.
\textit{J.-C. Bourin} [Int. J. Math. 20, No. 6, 679--691 (2009; Zbl 1181.15030)] raised the following question: let \(A\) and \(B\) be any two positive semi-definite matrices, and let \(t\in [0,1].\) It is true that \[ |\!|\!|A^t B^{1-t} + B^t A^{1-t} |\!|\!|\leq |\!|\!|A+B |\!|\!|? \] This question has been answered for the Hilbert-Schmidt norm under the condition \(t\in\left[\frac{1}{4}, \frac{3}{4}\right]\) (see [\textit{R. Bhatia}, J. Math. Phys. 55, No. 1, 013509, 3 p. (2014; Zbl 1288.15022); \textit{S. Hayajneh} and \textit{F. Kittaneh}, J. Math. Phys. 54, No. 3, 033504, 8 p. (2013; Zbl 1287.47011)]) and also for the trace norm (see [\textit{M. Hayajneh} et al., Int. J. Math. 28, No. 14, Article ID 1750102, 7 p. (2017; Zbl 1392.15031)]). Also a partial answer to this question in the wider class of unitarily invariant norms has been given in the above mentioned papers. In this paper, the authors give an affirmative answer to Bourin's question for \(t=\frac{1}{4}\) and \(\frac{3}{4}\).
For \(A\) and \(B\) \(n\times n\) positive semi-definite matrices, the authors prove, among other norm inequalities, that for \(t\in [0,1]\) and for all unitarily invariant norms \[ |\!|\!|A^t B^{1-t} + B^t A^{1-t} |\!|\!|\leq \sqrt{|\!|\!|A+B |\!|\!|\left(|\!|\!|A|\!|\!|+|\!|\!|B|\!|\!|\right)}. \] This norm inequality is stronger than the one obtained by Hayajneh and Kittaneh [loc. cit.].
Reviewer: Maria Graça (Lisboa)Extensions of some matrix inequalities via matrix meanshttps://www.zbmath.org/1475.150282022-01-14T13:23:02.489162Z"Yamazaki, Takeaki"https://www.zbmath.org/authors/?q=ai:yamazaki.takeaki
For the entire collection see [Zbl 1411.00046].Crystallographic groups, strictly tessellating polytopes, and analytic eigenfunctionshttps://www.zbmath.org/1475.200852022-01-14T13:23:02.489162Z"Rowlett, Julie"https://www.zbmath.org/authors/?q=ai:rowlett.julie"Blom, Max"https://www.zbmath.org/authors/?q=ai:blom.max"Nordell, Henrik"https://www.zbmath.org/authors/?q=ai:nordell.henrik"Thim, Oliver"https://www.zbmath.org/authors/?q=ai:thim.oliver"Vahnberg, Jack"https://www.zbmath.org/authors/?q=ai:vahnberg.jackThe authors generalize the results of \textit{P. H. Bérard} [Invent. Math. 58, 179--199 (1980; Zbl 0434.35068)] and \textit{B. J. McCartin} [Appl. Math. Sci., Ruse 2, No. 57--60, 2891--2901 (2008; Zbl 1187.35144)] to all dimensions. They prove that the following are equivalent: the first Dirichlet eigenfunction for the Laplace eigenvalue equation on a polytope is real analytic, the polytope strictly tessellates space, and the polytope is the fundamental domain of a crystallographic Coxeter group. They also show that under any of these equivalent conditions, all of the eigenfunctions are trigonometric functions. They connect these topics to the Fuglede and Goldbach conjectures and give a purely geometric formulation of Goldbach's conjecture.
Reviewer: Erich W. Ellers (Toronto)The equivalent parameter conditions for constructing multiple integral half-discrete Hilbert-type inequalities with a class of nonhomogeneous kernels and their applicationshttps://www.zbmath.org/1475.260212022-01-14T13:23:02.489162Z"He, Bing"https://www.zbmath.org/authors/?q=ai:he.bing.1"Hong, Yong"https://www.zbmath.org/authors/?q=ai:hong.yong"Chen, Qiang"https://www.zbmath.org/authors/?q=ai:chen.qiang.1Summary: In this paper, we establish equivalent parameter conditions for the validity of multiple integral half-discrete Hilbert-type inequalities with the nonhomogeneous kernel \(G(n^{\lambda_{1}}\|x\|_{m,\rho}^{\lambda_{2}})\) (\(\lambda_1, \lambda_2 >0\)) and obtain best constant factors of the inequalities in specific cases. In addition, we also discuss their applications in operator theory.More on operator Bellman inequalityhttps://www.zbmath.org/1475.260252022-01-14T13:23:02.489162Z"Mirzapour, F."https://www.zbmath.org/authors/?q=ai:mirzapour.farzollah"Morassaei, A."https://www.zbmath.org/authors/?q=ai:morassaei.ali"Moslehian, M. S."https://www.zbmath.org/authors/?q=ai:moslehian.mohammad-salSummary: We present a Bellman inequality involving operator means for operators acting on a Hilbert space. We also give some Bellman inequalities concerning sesquilinear forms. Finally, we refine the Jensen's operator inequality and use it for obtaining a refinement of the Bellman operator inequality.Coefficients of exponential series for analytic functions and the Pommiez operatorhttps://www.zbmath.org/1475.300152022-01-14T13:23:02.489162Z"Melikhov, S. N."https://www.zbmath.org/authors/?q=ai:melikhov.sergei-nSummary: In this paper, we present results of the existence of a linear continuous right inverse operator for the operator of the representation of analytic functions in a bounded convex domain of the complex plane by series of quasi-polynomials and exponentials. We also present closely related results on the A. F. Leontiev interpolating function and, more generally, on the interpolating functional and the corresponding Pommiez operator. We examine cyclic vectors and closed invariant subspaces of the Pommiez operator in weighted spaces of entire functions.On resolvent matrix, Dyukarev-Stieltjes parameters and orthogonal matrix polynomials via \([0, \infty)\)-Stieltjes transformed sequenceshttps://www.zbmath.org/1475.300852022-01-14T13:23:02.489162Z"Choque Rivero, A. E."https://www.zbmath.org/authors/?q=ai:choque-rivero.abdon-eddy"Mädler, C."https://www.zbmath.org/authors/?q=ai:madler.conradSummary: By using Schur transformed sequences and Dyukarev-Stieltjes parameters we obtain a new representation of the resolvent matrix corresponding to the truncated matricial Stieltjes moment problem. Explicit relations between orthogonal matrix polynomials and matrix polynomials of the second kind constructed from consecutive Schur transformed sequences are obtained. Additionally, a non-negative Hermitian measure for which the matrix polynomials of the second kind are the orthogonal matrix polynomials is found.Multiplicity of solutions to discrete inclusions with the \(p(k)\)-Laplace type equationshttps://www.zbmath.org/1475.352222022-01-14T13:23:02.489162Z"Ouaro, Stanislas"https://www.zbmath.org/authors/?q=ai:ouaro.stanislas"Zoungrana, Malick"https://www.zbmath.org/authors/?q=ai:zoungrana.malickThis paper is concerned with the existence and multiplicity of solutions to discrete inclusions with the \(p(k)\)-Laplace type equations.
They begin by giving some basic definitions and preliminary results where they define the generalized gradient of a function, coercive and anti-coercive function. Then they introduce three critical points theorem for locally Lipschitz functionals. With this theorem, the authors prove the existence of three \(m\)-periodic solutions and show that at least two of which are non-trivial.
For this, they define a functional \(J_m\) by
\[
J_{m}(u) = \sum_{k=1}^{m}A\big(k-1,\Delta u(k-1)\big) - \lambda \sum_{k=1}^{m}F\big(k,u(k)\big)
\]
and establish that a critical point of \(J_m\) is a solution of there equations.
By using three critical points theorem, they show that the problem has at least three solutions, at least two of which are necessarily non-zero.
Reviewer: Blaise Kone (Ouagadougou)Duality principles in Hilbert-Schmidt frame theoryhttps://www.zbmath.org/1475.420432022-01-14T13:23:02.489162Z"Dong, Jian"https://www.zbmath.org/authors/?q=ai:dong.jian"Li, Yun-Zhang"https://www.zbmath.org/authors/?q=ai:li.yunzhang.1|li.yunzhangThe concept of frame has several generalizations such as pseudo-frame, fusion frame (or frame of subspaces) and generalized frame (known as g-frame), among others. In particular, Hilbert-Schmidt (HS) frames are sequences of HS operators and are a more general version of g-frames. In this context, the notions of HS-Bessel sequences, HS-Riesz bases and HS-orthonormal bases are defined.
In the present paper, the notion of HS-R-dual sequence is introduced and used to establish some results about duality. It is proved that a sequence is an HS-frame (HS-frame sequence, HS-Riesz basis) if and only if its HS-R-dual sequence is an HS-Riesz sequence (HS-frame sequence, HS-Riesz basis) and the (unitary) equivalence between two HS-frames is characterized in terms of their HS-R-duals and transition matrices, respectively. HS-Rduals are characterized and it is proved that, given an HS-frame, among all its dual HS-frames, only the canonical dual admits minimal-norm HS-R-dual. Using HS-R-duals, dual HS-frame pairs are also characterized.
Reviewer: Patricia Mariela Morillas (San Luis)On the Volterra factorization of the Wiener-Hopf integral operatorhttps://www.zbmath.org/1475.450042022-01-14T13:23:02.489162Z"Arabadzhyan, L. G."https://www.zbmath.org/authors/?q=ai:arabadzhyan.l-gThe author studies a factorization problem to identify a classical Wiener-Hopf integral operator as the product of upper and lower Volterra integral operators. The aim is to use this result to illustrate how to represent solutions of first- and second-kind Wiener-Hopf equations.
Reviewer: Luis Filipe Pinheiro de Castro (Aveiro)On the relationship between the factorization problem in the Wiener algebra and the truncated Wiener-Hopf equationhttps://www.zbmath.org/1475.450072022-01-14T13:23:02.489162Z"Voronin, A. F."https://www.zbmath.org/authors/?q=ai:voronin.anatolii-fedorovichThe author studies a homogeneous vector Riemann boundary value problem (i.e., a factorization problem) from a new point of view. Namely, he reduces the Riemann problem of right factorization of a second order \(2\times2\) matrix function from the Wiener algebra \(\mathcal{G}\mathbb{W}^{2\times2}(\mathbb{R})\), that is \[M(x)=\Psi_+(x)D(x)\Psi_-^{-1}(x),\]
\[M(x)=\left[\begin{array}{cc}1&m^+(x)\\
m^-(x)&1+m^-(x)m^+(x)\end{array}\right], \] to the solution of a truncated Wiener-Hopf equation (i.e., a convolution equation in a finite interval). For this connection the author needs further constraints \[ m^\pm(x)=m^\pm_\tau(x)=\int_{-\tau}^\tau e^{ixt}\mu_\pm(t)dt,\qquad x\in\mathbb{R},\] where
\[
\mu_\pm(t)=\theta(\pm t)\mu(t),\qquad \mu\in L(\mathbb{R}), \qquad \tau>0.
\]
If
\[
d^\pm(x)=1\pm e^{\pm ix\tau}m^\pm_\tau(x),\]
\[ \hat{k}(x)=\frac1{d^-(x)d^+(x)}-1,\qquad k(x)=(\mathcal{F}^{-1}\hat{k})(x),
\]
the truncated Wiener-Hopf equation \[ u(t) + \int_0^\tau k(t-s)u(s)ds=f(t),\qquad t\in[0,\tau],\] where the right-hand side \(f(t)\) is constructed from the data of the factorable matrix, is associated with the above factorization problem. It is proved also that if the truncated homogeneous Wiener-Hopf equation has only the trivial solution in \(L_1(0,\tau)\), then the partial indices of the matrix \(M(x)\) are zero.
Reviewer: Roland Duduchava (Tbilisi)On a generalization of a complementary triangle inequality in Hilbert spaces and Banach spaceshttps://www.zbmath.org/1475.460142022-01-14T13:23:02.489162Z"Sain, Debmalya"https://www.zbmath.org/authors/?q=ai:sain.debmalyaSummary: We study a possible generalization of a complementary triangle inequality in Hilbert spaces and Banach spaces. Our results in the present article improve and generalize some of the earlier results in this context. We also present an operator norm inequality in the setting of Banach spaces, as an application of the present study.Characterizations of norm-parallelism in spaces of continuous functionshttps://www.zbmath.org/1475.460152022-01-14T13:23:02.489162Z"Zamani, Ali"https://www.zbmath.org/authors/?q=ai:zamani.ali|zamani.ali-rezaSummary: In this paper, we consider the characterization of norm-parallelism problem in some classical Banach spaces. In particular, for two continuous functions \(f\), \(g\) on a compact Hausdorff space \(K\), we show that \(f\) is norm-parallel to \(g\) if and only if there exists a probability measure (i.e., positive and of full measure equal to 1) \(\mu \) with its support contained in the norm-attaining set \(\{x\in K: \, |f(x)| = \Vert f\Vert \}\) such that \(\big |\int_K \overline{f(x)}g(x)\mathrm{d}\mu (x)\big | = \Vert f\Vert \,\Vert g\Vert \).Essential normality for Beurling-type quotient modules over tube-type domainshttps://www.zbmath.org/1475.460232022-01-14T13:23:02.489162Z"Zhang, Shuyi"https://www.zbmath.org/authors/?q=ai:zhang.shuyiSummary: In this note we investigate the essential normality of the Beurling-type quotient module \({\mathcal{D}}:=H^2(\Omega)\ominus \eta H^2(\Omega)\) with an inner function \(\eta\) inside \(A(\Omega)\) over an irreducible tube-type domain \(\Omega\). For the Lie ball (of rank 2), we characterize the essential normality of the corresponding quotient Hardy module and determine its essential spectrum. For domains of higher rank, we introduce the analogous concept of \(k\)-normality and again characterize \((r-1)\)-normality in terms of representation theory of the maximal compact subgroup.On 2-inner product spaces and reproducing propertyhttps://www.zbmath.org/1475.460252022-01-14T13:23:02.489162Z"Sababe, Saeed Hashemi"https://www.zbmath.org/authors/?q=ai:hashemi-sababe.saeedSummary: This paper is devoted to study the reproducing property on 2-inner product Hilbert spaces. We focus on a new structure to produce reproducing kernel Hilbert and Banach spaces. According to multi variable computing, these structures play a key role in probability, mathematical finance, and machine learning.Mapping properties of operator-valued pseudo-differential operatorshttps://www.zbmath.org/1475.460532022-01-14T13:23:02.489162Z"Xia, Runlian"https://www.zbmath.org/authors/?q=ai:xia.runlian"Xiong, Xiao"https://www.zbmath.org/authors/?q=ai:xiong.xiaoThe authors investigate pseudo-differential operators \(T^c_\sigma\) given by a symbol \(\sigma\) defined on \(\mathbb{R}^d\times\mathbb{R}^d\) with values in a von Neumann algebra \(\mathcal{M}\) by setting \(T^c_\sigma f(s)=\int_{\mathbb{R}^d}\sigma(s,\xi)\hat f(\xi)e^{2\pi\mathrm{i}s\cdot\xi}\,d\xi\), where \(f\) is defined on \(\mathbb{R}^d\) with values in the space \(L_1(\mathcal{M})+\mathcal{M}\). Note that when \(\mathcal{M}\) is noncommutative, setting \(T^r_\sigma f(s)=\int_{\mathbb{R}^d}\hat f(\xi)\sigma(s,\xi)e^{2\pi\mathrm{i}s\cdot\xi}\,d\xi\) defines a different operator: the superscripts ``\(r\)'' and ``\(c\)'' stand respectively for ``row'' and ``column''. The hypothesis on the symbol is that it belong to the \textit{Hörmander class} \(S^0_{1,\delta}\), i.e., that
\[
\|D_s^\gamma D_\xi^\beta\sigma(s,\xi)\|_{\mathcal{M}}\le C_{\gamma,\beta}(1+|\xi|)^{\delta(\gamma_1+\cdots+\gamma_d)-\beta_1-\cdots-\beta_d}
\]
for all multi-indices \(\gamma,\beta\) of nonnegative integers. Their techniques and results differ according to whether \(0\le\delta<1\) (the case of ``regular'' symbols) or \(\delta=1\) (``forbidden'' symbols). The operators \(T^c_\sigma\) are proved to be bounded on the following three kinds of spaces: (i) the operator-valued Sobolev spaces \(H_2^\alpha(\mathbb{R}^d;L_2(\mathcal{M}))\); (ii) the operator-valued Besov spaces \(B_{p,q}^\alpha(\mathbb{R}^d;L_p(\mathcal{M}))\); (iii) the operator-valued column inhomogeneous Triebel-Lizorkin spaces \(F_{p}^{\alpha,c}(\mathbb{R}^d;\mathcal{M})\). Here, \(\alpha\) is an arbitrary real number if \(\sigma\) is regular and \(\alpha>0\) if \(\sigma\) is forbidden. Case (iii) has been introduced by the authors in
[Integral Equations Oper. Theory 90, No. 6, Paper No. 65, 65~p. (2018; Zbl 06965992)]:
they use decompositions of \(f\) into smooth atoms obtained there and study the action of \(T^c_\sigma\) on those.
At the end of the article, they obtain the same results for tori \(\mathbb{T}^d\) and quantum tori \(\mathbb{T}_\theta^d\) instead of \(\mathbb{R}^d\). They also discuss how their article relates to
[\textit{A.~González-Pérez}, \textit{M.~Junge} and \textit{J.~Parcet}, Singular integrals in quantum Euclidean spaces. Mem. Amer. Math. Soc. 272, No.~1334, Providence, RI: American Mathematical Society (2021)].
Reviewer: Stefan Neuwirth (Besançon)Quaternionic de Branges spaces and characteristic operator functionhttps://www.zbmath.org/1475.470012022-01-14T13:23:02.489162Z"Alpay, Daniel"https://www.zbmath.org/authors/?q=ai:alpay.daniel"Colombo, Fabrizio"https://www.zbmath.org/authors/?q=ai:colombo.fabrizio"Sabadini, Irene"https://www.zbmath.org/authors/?q=ai:sabadini.ireneIn this book, the authors present the study of the de Branges spaces in the quaternionic setting, with a focus on the notion of characteristic operator function of a bounded linear operator with finite real part, and address several important questions such as the study of \(J\)-contractions (here, \(J\) is a self-adjoint, unitary operator) and the inverse problem, that is, to characterize which \(J\)-contractive functions are characteristic functions of an operator. In particular, they prove Potapov's factorization theorem. The details of the topics covered in this book are described below.
In Chapter~1, details about the characteristic operator function of a linear operator defined on \(\mathbb C^n\) and taking values in a Hilbert space is discussed and later the same is discussed in the quaternionic setting. In Chapter~2, quaternionic matrices and the associated complex matrices, quaternionic Toeplitz and Hankel matrices, and quaternionic Krein spaces are studied. The relations between the quaternionic reproducing kernel Pontryagin spaces and Hermitian kernels are also discussed in the same chapter. Chapter~3 is dedicated to the study of slice hyperholomorphic functions, the \(S\)-resolvent operators and the \(S\)-spectrum, and \(S\)-functional calculus for the left slice hyperholomorphic functions. In Chapters 4 and 5, Wiener-Hopf factorization and Rota's model for linear operators are considered in the quaternionic setting.
In Chapter 6, the space \(\mathcal{H}(A,B)\) is considered and the authors provide counterparts of various results in this framework, including the operator multiplication in the half-space case and in the unit ball case, and the study of reproducing kernels. The case of \(J\)-contractive functions is presented in Chapter~7.
In Chapter 8, the characteristic operator function in the quaternionic setting is discussed. Some classes of functions with a positive real part in the half-space or the unit ball are discussed in Chapter~9. The final Chapter~10 is devoted to the canonical differential systems in the quaternionic setting.
Reviewer: G. Ramesh (Sangareddy)Jeribi essential spectrumhttps://www.zbmath.org/1475.470032022-01-14T13:23:02.489162Z"Belabbaci, Chafika"https://www.zbmath.org/authors/?q=ai:belabbaci.chafikaSummary: The aim of this paper is to present some results concerning the Jeribi essential spectrum. We use the notion of measure of weak noncompactness to give a formulae for the Jeribi essential spectral radius and we use the class of Tauberian and cotauberian operators to present some relationship between the Jeribi essential spectrum and the other essential spectra.A generalization of the numerical radiushttps://www.zbmath.org/1475.470042022-01-14T13:23:02.489162Z"Abu-Omar, Amer"https://www.zbmath.org/authors/?q=ai:abu-omar.amer"Kittaneh, Fuad"https://www.zbmath.org/authors/?q=ai:kittaneh.fuadSummary: We define a norm on the space of bounded linear operators on a Hilbert space, which generalizes the numerical radius norm. We investigate basic properties of this norm and prove inequalities involving it. A concrete example of this norm is also given.Numerical radius inequalities and its applications in estimation of zeros of polynomialshttps://www.zbmath.org/1475.470052022-01-14T13:23:02.489162Z"Bhunia, Pintu"https://www.zbmath.org/authors/?q=ai:bhunia.pintu"Bag, Santanu"https://www.zbmath.org/authors/?q=ai:bag.santanu"Paul, Kallol"https://www.zbmath.org/authors/?q=ai:paul.kallolThe authors give an upper bound and a lower bound for the numerical radius of a bounded linear operator $T$ acting on a Hilbert space, which improves the existing bounds given in [\textit{A. Abu-Omar} and \textit{F. Kittaneh}, Ann. Funct. Anal. 5, No. 1, 56--62 (2014; Zbl 1298.47011)] and [\textit{F. Kittaneh}, Stud. Math. 168, No. 1, 73--80 (2005; Zbl 1072.47004)], respectively. Moreover, they present an upper bound of the numerical radius in terms of $\Vert\Re(e^{i\theta}T)\Vert$ and a lower bound of the numerical radius in terms of the spectral values of the real part $\Re(T)$ and the imaginary part $\Im(T)$ of $T$. In addition, they estimate the spectral radius of sum of product of $n$ pairs of operators. As an application, they estimate the zeros of a polynomial.
Reviewer: Mohammad Sal Moslehian (Mashhad)Sequences of bounds for the spectral radius of a positive operatorhttps://www.zbmath.org/1475.470062022-01-14T13:23:02.489162Z"Drnovšek, Roman"https://www.zbmath.org/authors/?q=ai:drnovsek.romanSummary: In 1992, Szyld provided a sequence of lower bounds for the spectral radius of a nonnegative matrix \(A\), based on the geometric symmetrization of powers of \(A\) [\textit{D. B. Szyld}, ibid. 174, 239--242 (1992; Zbl 0758.15013)]. In [ibid. 273, 23--28 (1998; Zbl 0901.15012)], \textit{D. Taşçi} and \textit{S. Kirkland} proved a companion result by giving a sequence of upper bounds for the spectral radius of \(A\), based on the arithmetic symmetrization of powers of \(A\). In this note, we extend both results to positive operators on \(L^2\)-spaces.Isometric dilations and von Neumann inequality for a class of tuples in the polydischttps://www.zbmath.org/1475.470072022-01-14T13:23:02.489162Z"Barik, Sibaprasad"https://www.zbmath.org/authors/?q=ai:barik.sibaprasad"Das, B. Krishna"https://www.zbmath.org/authors/?q=ai:krishna-das.b"Haria, Kalpesh J."https://www.zbmath.org/authors/?q=ai:haria.kalpesh-j"Sarkar, Jaydeb"https://www.zbmath.org/authors/?q=ai:sarkar.jaydebSummary: The celebrated Sz.-Nagy and Foias and Ando theorems state that a single contraction, or a pair of commuting contractions, acting on a Hilbert space always possesses isometric dilation and subsequently satisfies the von Neumann inequality for polynomials in \( \mathbb{C}[z]\) or \( \mathbb{C}[z_1, z_2]\), respectively. However, in general, neither the existence of isometric dilation nor the von Neumann inequality holds for \( n\)-tuples, \( n \geq 3\), of commuting contractions. The goal of this paper is to provide a taste of isometric dilations, von Neumann inequality, and a refined version of von Neumann inequality for a large class of \( n\)-tuples, \( n \geq 3\), of commuting contractions.Applications of fixed point theorems in the theory of invariant subspaceshttps://www.zbmath.org/1475.470082022-01-14T13:23:02.489162Z"Espínola, Rafa"https://www.zbmath.org/authors/?q=ai:espinola-garcia.rafael"Lacruz, Miguel"https://www.zbmath.org/authors/?q=ai:lacruz.miguelSummary: We survey several applications of fixed point theorems in the theory of invariant subspaces. The general idea is that a fixed point theorem applied to a suitable map yields the existence of invariant subspaces for an operator on a Banach space.Cyclicity in Dirichlet spaceshttps://www.zbmath.org/1475.470092022-01-14T13:23:02.489162Z"Elmadani, Y."https://www.zbmath.org/authors/?q=ai:elmadani.y"Labghail, I."https://www.zbmath.org/authors/?q=ai:labghail.iThis article presents several results about cyclic elements of the harmonically weighted Dirichlet space \(\mathcal{D}(\mu)\) induced by a positive finite Borel measure \(\mu\) on the unit circle. The author extends a theorem of \textit{L. Brown} and \textit{W. Cohn} [Proc. Am. Math. Soc. 95, 42--46 (1985; Zbl 0597.30064)], and shows that, if \(E\) is a closed subset of the unit disc with zero \(c_{\mu}\) capacity, then there exists a function \(f \in \mathcal{D}(\mu)\) which is cyclic, vanishes on \(E\), and is uniformly continuous. Moreover, a sufficient condition for a continuous function on the closed unit disk to be cyclic in \(\mathcal{D}(\mu)\) is provided.
Reviewer: José Bonet (Valencia)A hypercyclicity criterion for non-metrizable topological vector spaceshttps://www.zbmath.org/1475.470102022-01-14T13:23:02.489162Z"Peris, Alfred"https://www.zbmath.org/authors/?q=ai:peris.alfredoSummary: We provide a sufficient condition for an operator \(T\) on a non-metrizable and sequentially separable topological vector space \(X\) to be sequentially hypercyclic. This condition is applied to some particular examples, namely, a composition operator on the space of real analytic functions on $\,]0,1[\,$, which solves two problems of \textit{J. Bonet} and \textit{P. Domański} [Math. Proc. Camb. Philos. Soc. 153, No. 3, 489--503 (2012; Zbl 1272.47037)], and the ``snake shift'' constructed in [\textit{J. Bonet} et al., Bull. Lond. Math. Soc. 37, No. 2, 254--264 (2005; Zbl 1150.47005)] on direct sums of sequence spaces. The two examples have in common that they do not admit a densely embedded F-space \(Y\) for which the operator restricted to \(Y\) is continuous and hypercyclic, i.e., the hypercyclicity of these operators cannot be a consequence of the comparison principle with hypercyclic operators on F-spaces.Shape, scale, and minimality of matrix rangeshttps://www.zbmath.org/1475.470112022-01-14T13:23:02.489162Z"Passer, Benjamin"https://www.zbmath.org/authors/?q=ai:passer.benjamin-wAuthor's abstract: We study containment and uniqueness problems concerning matrix convex sets. First, to what extent is a matrix convex set determined by its first level? Our results in this direction quantify the disparity between two product operations, namely, the product of the smallest matrix convex sets over \(K_i \subseteq \mathbb{C}^d\) and the smallest matrix convex set over the product of \(K_i\). Second, if a matrix convex set is given as the matrix range of an operator tuple \(T\), when is \(T\) determined uniquely? We provide counterexamples to results in the literature, showing that a compact tuple meeting a minimality condition need not be determined uniquely, even if its matrix range is a particularly friendly set. Finally, our results may be used to improve dilation scales, such as the norm bound on the dilation of (not necessarily self-adjoint) contractions to commuting normal operators, both concretely and abstractly.
Reviewer: Jaydeb Sarkar (Bangalore)The fundamental equations for the generalized resolvent of an elementary pencil in a unital Banach algebrahttps://www.zbmath.org/1475.470122022-01-14T13:23:02.489162Z"Albrecht, Amie"https://www.zbmath.org/authors/?q=ai:albrecht.amie-r"Howlett, Phil"https://www.zbmath.org/authors/?q=ai:howlett.phil-g"Verma, Geetika"https://www.zbmath.org/authors/?q=ai:verma.geetikaSummary: We show that the generalized resolvent of a linear pencil in a unital Banach algebra over the field of complex numbers is analytic on an open annular region of the complex plane if and only if the coefficients of the Laurent series expansion satisfy a system of left and right fundamental equations and are geometrically bounded. Our analysis includes the case where the resolvent has an isolated essential singularity at the origin. We find a closed form for the resolvent and use the fundamental equations to establish key spectral separation properties when the resolvent has only a finite number of isolated singularities. We show that our results can be used to solve an infinite system of ordinary differential equations and to solve the generalized Sylvester equation. We also show that our results can be extended to polynomial pencils.On some inequalities with operators in Hilbert spaceshttps://www.zbmath.org/1475.470132022-01-14T13:23:02.489162Z"Cărăuşu, Alexandru"https://www.zbmath.org/authors/?q=ai:carausu.alexandruSummary: Operators defined on Hilbert spaces represent a major subfield of (or base for) the functional analysis. Several types of inequalities among such operators were established and studied in the last decades, mainly by the early 1950s and then in the 1980s. In this paper there are reviewed some of the most important types of inequalities, introduced and studied by H. Bohr, E. Heinz and T. Kato, H. Weyl, W. Reid, and other authors. They were extended and/or sharpened by other authors, mentioned in the Introduction. Some of the definitions and proofs, found in several references, are completed (by the author) with specific formulas from Hilbert space theory, some details are also added to certain proofs and definitions as well. The main ways for establishing inequalities with operators are pointed out: scalar inequalities like the Cauchy-Schwarz and Bohr's inequalities over the complex field or on an H-space, certain identities with H-space operators, etc.Refinements and reverses for the relative operator entropy $S (A | B)$ when $B \ge A$https://www.zbmath.org/1475.470142022-01-14T13:23:02.489162Z"Dragomir, S. S."https://www.zbmath.org/authors/?q=ai:dragomir.sever-silvestru"Buşe, C."https://www.zbmath.org/authors/?q=ai:buse.constantinSummary: In this paper we obtain new refinements and reverse inequalities for the relative operator entropy \(S (A|B)\) of two positive invertible operators when \(B \geq A\). Applications for the operator entropy \(\eta(C)\) in the case of positive contractions \(C\) are also given.On (\(A, m\))-symmetric operators in a Hilbert spacehttps://www.zbmath.org/1475.470152022-01-14T13:23:02.489162Z"Jeridi, N."https://www.zbmath.org/authors/?q=ai:jeridi.n"Rabaoui, R."https://www.zbmath.org/authors/?q=ai:rabaoui.rchidSummary: For a positive \(A\in \mathcal {B}(\mathbb {H})\), an operator \(T\in \mathcal {B}(\mathbb {H})\) is said to be (\(A\), \(m\))-symmetric if it satisfies the operator equation \(\sum \nolimits _{k=0}^{m}(-1)^{m-k} \binom mk T^{*k}AT^{m-k}=0.\) This class of operators seems a natural generalization of \(m\)-symmetric operators on a Hilbert space. In this paper, first we give various properties related to such a family. Then, we prove that, if \(T\) and \(Q\) are commuting operators, \(T\) is (\(A\), \(m\))-symmetric and \(Q\) is \(l\)-nilpotent, then \((T+Q)\) is \((A,m + 2l - 2)\)-symmetric. In addition, we show that every power of an (\(A\), \(m\))-symmetric operator is also (\(A\), \(m\))-symmetric. Some connection between (\(A\), \(m\))-symmetric operators and \(C_0\)-semigroups are also shown. Finally, we characterize the spectra of such operators.Curvature inequalities for operators in the Cowen-Douglas class and localization of the Wallach sethttps://www.zbmath.org/1475.470172022-01-14T13:23:02.489162Z"Misra, Gadadhar"https://www.zbmath.org/authors/?q=ai:misra.gadadhar"Pal, Avijit"https://www.zbmath.org/authors/?q=ai:pal.avijitLet \(\Omega\) be a bounded domain in \(\mathbb{C}^m\). The Cowen-Douglas class consists of \(m\)-tuples of commuting bounded operators \(\mathbf{T}=(T_1,\dots,T_m)\) on a Hilbert space \(\mathcal{H}\) such that for \(w=(w_1,\dots,w_m)\in\Omega\), the dimension of the joint kernel \(\cap_{k=1}^m\ker(T_k-w_kI)\) is equal to \(1\); for \(w\in\Omega\) and \(h\in\mathcal{H}\), the operator \(D_{\mathbf{T}-wI}:\mathcal{H}\to\mathcal{H}\oplus\dots\oplus\mathcal{H}\) defined by \(D_{\mathbf{T}-wI}=((T_1-w_1I)h,\dots,(T_m-w_mI)h))\) has closed range; the closed linear span of \(\left\{\cap_{k=1}^m\ker (T_k-w_kI):w\in\Omega\right\}\) coincides with \(\mathcal{H}\). For an \(m\)-tuple \(\mathbf{T}\in B_1(\Omega)\), let \(N_{\mathbf{T}}(w)\) denote the restriction of \(\mathbf{T}\) to the subspace \(\mathcal{N}(w):=\cap_{i,j=1}^m\ker (T_i-w_iI)(T_j-w_jI)\). The authors study the contractivity and complete contractivity of homomorphisms of the polynomial ring \(P[z_1,\dots,z_m]\) defined by \(\rho_{N_{\mathbf{T}}(w)}(p)=p(N_{\mathbf{T}}(w))\) for \(p\in P[z_1,\dots,z_m]\). Some explicit examples arising from multiplication operators on the Bergman space of \(\Omega\) are considered in detail. For operators in \(B_1(\Omega)\), there exists a holomorphic choice \(\gamma(w)\) of eigenvalues at \(w\). The map \(\gamma\) defines a holomorphic Hermitian vector bundle \(E_{\mathbf{T}}\) on \(\Omega\). One can define the curvature \(\mathcal{K}_{\mathbf{T}}(w):=-\left(\left( \frac{\partial^2}{\partial w_i\partial\overline{w_j}}\log\|\gamma(w)\|^2 \right)\right)_{i,j=1}^m\). It is shown that contractive properties of \(\rho_{N_{\mathbf{T}}(w)}\) are equivalent to an inequality for the curvature of the Cowen-Dougles bundle \(E_{\mathbf{T}}\).
Reviewer: Alexei Yu. Karlovich (Lisboa)On \((m,\infty )\)-isometries: exampleshttps://www.zbmath.org/1475.470212022-01-14T13:23:02.489162Z"Bermúdez, Teresa"https://www.zbmath.org/authors/?q=ai:bermudez.teresa"Zaway, Hajer"https://www.zbmath.org/authors/?q=ai:zaway.hajerSummary: An operator \(T\) on a Banach space \(X\) is said to be an \((m,\infty )\)-isometry if \[\max_{\substack{0 \leq k \leq m\\ k \;\; \text{even}}} || T^kx|| =\max_{\substack {0 \leq k \le m\\ k \;\; \text{odd}}}||T^kx||,\] for all \(x\in X\). In this paper, we study unilateral weighted shift operators which are \((m,\infty )\)-isometries for some integers \(m\). In particular, we show that any power of an \((m,\infty )\)-isometry is not necessarily an \((m,\infty )\)-isometry. We also study strict \((3,\infty )\)-isometries on \(\mathbb {R}^2\) and give an example of a strict \((2n-1, \infty )\)-isometry on \({\mathbb {C}}^2\), for any odd integer \(n\).On Rubel type operator equations on the space of analytic functionshttps://www.zbmath.org/1475.470222022-01-14T13:23:02.489162Z"Linchuk, Yuriy S."https://www.zbmath.org/authors/?q=ai:linchuk.yu-sSummary: The purpose of this work is to describe all pairs of linear operators that act in spaces of analytic functions in domains and satisfy the Rubel's type operator equations. We describe all generalized derivation pairs of linear operators on the space of functions analytic in domains. Using this result we describe all pairs of linear operators that act in the spaces of functions analytic in domains and satisfying the operator analog of Kannappan-Nandakumar's equation [\textit{P. Kannappan} and \textit{N. R. Nandakumar}, Aequationes Math. 61, No. 3, 233--238 (2001; Zbl 0990.39022)].The free Grothendieck theoremhttps://www.zbmath.org/1475.470992022-01-14T13:23:02.489162Z"Augat, Meric L."https://www.zbmath.org/authors/?q=ai:augat.meric-lThe author proves the free Grothendieck theorem on bijective polynomial mappings of \(\mathbb{C}^g\). The approach involves careful analysis of the noncommutative Jacobian matrix. Indeed, the author shows that, if \(p\) is a polynomial mapping in \(g\) freely non-commuting variables sending \(g\)-tuples of matrices to \(g\)-tuples of matrices that is injective, then it has a free polynomial inverse.
Reviewer: Ghadir Sadeghi (Sabzevār)Regularity of shape optimizers for some spectral fractional problemshttps://www.zbmath.org/1475.490522022-01-14T13:23:02.489162Z"Tortone, Giorgio"https://www.zbmath.org/authors/?q=ai:tortone.giorgioSummary: This paper is dedicated to the spectral optimization problem
\[ \min \{ \lambda_1^s ( \Omega ) + \cdots + \lambda_m^s ( \Omega ) + \Lambda \mathcal{L}_n ( \Omega ) : \Omega \subset D \text{ s-quasi-open} \}\]
where \(\Lambda > 0\), \(D \subset \mathbb{R}^n\) is a bounded open set and \(\lambda_i^s(\Omega)\) is the \(i\)-th eigenvalue of the fractional Laplacian on \(\Omega\) with Dirichlet boundary condition on \(\mathbb{R}^n \smallsetminus \Omega\). We first prove that the first \(m\) eigenfunctions on an optimal set are locally Hölder continuous in the class \(C^{0 , s}\) and, as a consequence, that the optimal sets are open sets. Then, via a blow-up analysis based on a Weiss type monotonicity formula, we prove that the topological boundary in \(D\) of a minimizer \(\Omega\) is composed of a relatively open \textit{regular part} and a closed \textit{singular part} of Hausdorff dimension at most \(n - n^\ast \), for some \(n^\ast \geq 3\). Finally we use a viscosity approach to prove \(C^{1 , \alpha} \)-regularity of the regular part of the boundary.Partial retraction note to: ``Two-term, asymptotically sharp estimates for eigenvalue means of the Laplacian''https://www.zbmath.org/1475.580212022-01-14T13:23:02.489162Z"Harrell, Evans M. II"https://www.zbmath.org/authors/?q=ai:harrell.evans-m-ii"Stubbe, Joachim"https://www.zbmath.org/authors/?q=ai:stubbe.joachimFrom the text: We regret that we have to retract portions of our article [ibid. 8, No. 4, 1529--1550 (2018; Zbl 1402.58020)] due to an essential error in the proof of Theorem 1.2, which is used in other places in the paper.Formulae for the derivative of the Poincaré constant of Gibbs measureshttps://www.zbmath.org/1475.600942022-01-14T13:23:02.489162Z"Sieber, Julian"https://www.zbmath.org/authors/?q=ai:sieber.julianSummary: We establish formulae for the derivative of the Poincaré constant of Gibbs measures on both compact domains and all of \(\mathbb{R}^d\). As an application, we show that if the (not necessarily convex) Hamiltonian is an increasing function, then the Poincaré constant is strictly decreasing in the inverse temperature, and vice versa. Applying this result to the \(O(2)\) model allows us to give a sharpened upper bound on its Poincaré constant. We further show that this model exhibits a qualitatively different zero-temperature behavior of the Poincaré and Log-Sobolev constants.Derivative free regularization method for nonlinear ill-posed equations in Hilbert scaleshttps://www.zbmath.org/1475.650352022-01-14T13:23:02.489162Z"George, Santhosh"https://www.zbmath.org/authors/?q=ai:george.santhosh"Kanagaraj, K."https://www.zbmath.org/authors/?q=ai:kanagaraj.kSummary: In this paper, we deal with nonlinear ill-posed operator equations involving a monotone operator in the setting of Hilbert scales. Our convergence analysis of the proposed derivative-free method is based on the simple property of the norm of a self-adjoint operator. Using a general Hölder-type source condition, we obtain an optimal order error estimate. Also, we consider the adaptive parameter choice strategy proposed by \textit{S. Pereverzev} and \textit{E. Schock} [SIAM J. Numer. Anal. 43, No. 5, 2060--2076 (2005; Zbl 1103.65058)] for choosing the regularization parameter. Finally, we apply the proposed method to the parameter identification problem in an elliptic PDE in the setting of Hilbert scales and compare the results with the corresponding method in Hilbert space.A Birman-Schwinger principle in galactic dynamicshttps://www.zbmath.org/1475.850022022-01-14T13:23:02.489162Z"Kunze, Markus"https://www.zbmath.org/authors/?q=ai:kunze.markus-christian|kunze.markusOne of the simplest versions of the Birman-Schwinger principle is that a negative real number \(-e\) is an eigenvalue of the Schrödinger operator \(H:=-\Delta+V\) acting in \(L^2(\mathbb{R}^n)\) with the potential \(V\le 0\) if and only if \(1\) is an eigenvalue of the operator \(B_e:=\sqrt{-V}(-\Delta+e)^{-1}\sqrt{-V}\), and there are moreover simple formulas that relate the eigenfunctions of \(H\) and \(B_e\), respectively. The usefulness of this approach is due to the fact that the operator \(B_e\) is a nonnegative Hilbert-Schmidt operator under suitable decay hypothesies on the potential~\(V\). The book under review develops a version of the Birman-Schwinger principle which is applicable in certain stability problems of the non-relativistic galactic dynamics. As mentioned in the introductory chapter, the time evolution of self-gravitating matter is described by the Vlasov-Poisson system \(\partial_tf(t,x,v)+v\cdot\nabla_x f(t,x,v)-\nabla_xU_f(t,x)\cdot\nabla_vf(t,x,v)=0\), where \((t,x,v)\in \mathbb{R}\times\mathbb{R}^3\times\mathbb{R}^3\) and \(U_f(t,x)=-\int_{\mathbb{R}^3}\frac{\rho_f(t,y)}{\vert y-x\vert}dy\), while \(\rho_f(t,y)=\int_{\mathbb{R}^3}f(t,y,\cdot)\). Let \(Q=Q(x,v)\) be a steady state solution with the particle energy \(e_Q(x,v)=\frac{1}{2}\vert v\vert^2+U_Q(x)\), and define its corresponding operators \(g\mapsto \mathcal{T}g:=\{g,e_Q\}\) and \(g\mapsto \mathcal{K}g:=\{Q,U_g\}\), where \(\{\cdot,\cdot\}\) is the canonical Poisson bracket on \(\mathbb{R}^3\times\mathbb{R}^3\). One then proves that, under appropriate hypotheses on the steady state solution~\(Q\), the operator \(u\mapsto Lu:=-\mathcal{T}^2u-\mathcal{K}\mathcal{T}u\) is essentially selfadjoint and bounded from below with respect to a suitable inner product depending on~\(Q\) (Theorem 1.2). It is to that operator \(L\) that one associates a family of non-negative Hilbert-Schmidt operators \(\mathcal{Q}_\lambda\) with the properties similar to the operators \(B_e\) associated to the Schrödinger operator~\(H\). Using this approach, one can determine when the best constant in the Anosov stability estimate is attained (i.e., the aforementioned lower boundedness property of~\(L\)). The main body of the monograph contains the rigorous development of that approach, and there are also several appendices where the most technical details are treated.
Reviewer: Daniel Beltiţă (Bucureşti)