Recent zbMATH articles in MSC 47Ahttps://zbmath.org/atom/cc/47A2021-02-27T13:50:00+00:00WerkzeugOperator inequalities related to angular distances.https://zbmath.org/1453.260322021-02-27T13:50:00+00:00"Taba, Davood Afkhami"https://zbmath.org/authors/?q=ai:taba.davood-afkhami"Dehghan, Hossein"https://zbmath.org/authors/?q=ai:dehghan.hosseinSummary: For any nonzero elements \(x,y\) in a normed space \(X\), the angular and skew-angular distance is respectively defined by \(\alpha[x,y]=\left\|\frac x{\|x\|}-\frac y{\|y\|}\right\|\) and \(\beta[x,y]=\left\|\frac x{\|y\|}-\frac y{\|x\|}\right\|\). Also inequality \(\alpha\leq\beta\) characterizes inner product spaces. Operator version of \(\alpha\) has been studied by \textit{J. Pečarić} and \textit{R. Rajić} [J. Math. Inequal. 4, No. 1, 1--10 (2010; Zbl 1186.26020)], \textit{K.-S. Saito} and \textit{M. Tominaga} [Linear Algebra Appl. 432, No. 12, 3258--3264 (2010; Zbl 1195.26044)], and \textit{L. Zou} et al. [Linear Algebra Appl. 438, No. 1, 436--442 (2013; Zbl 1267.47028)]. In this paper, we study the operator version of \(\beta\) by using Douglas' lemma. We also prove that the operator version of inequality \(\alpha\leq\beta\) holds for commutating normal operators. Some examples are presented to show essentiality of these conditions.Fredholm property of integral operators with homogeneous kernels of compact type in the \(L_2\) space on the Heisenberg group.https://zbmath.org/1453.430032021-02-27T13:50:00+00:00"Denisenko, V. V."https://zbmath.org/authors/?q=ai:denisenko.valerii-vasilevich"Deundyak, V. M."https://zbmath.org/authors/?q=ai:deundyak.v-mLet \(\mathbb{H}^n\) be the Heisenberg group. There is a one-parameter group of dilations \(\delta(t)\), which act as automorphisms of \(\mathbb{H}^n\). There is a steady interest in operators generated by convolutions on \(\mathbb{H}^n\) with \(\delta(t)\)-homogeneous kernels, which were investigated starting from [\textit{A. S. Dynin}, Sov. Math., Dokl. 16, 1608--1612 (1975; Zbl 0328.58017); translation from Dokl. Akad. Nauk SSSR 225, 1245--1248 (1975); Sov. Math., Dokl. 17, 508--512 (1976; Zbl 0338.35086); translation from Dokl. Akad. Nauk SSSR 227, 792--795 (1976)]. The paper under review introduces the unital \(C^*\)-algebra generated by integral operators with \(\delta(t)\)-homogeneous kernels of compact type and multiplicatively weakly oscillating coefficients. Following Simonenko's profound localisation technique [\textit{I. B. Simonenko}, Izv. Akad. Nauk SSSR, Ser. Mat. 29, 567--586, 757--782 (1965; Zbl 0146.13101)] the authors construct a symbolic calculus. There are necessary and sufficient conditions for an operator to have the Fredholm property in terms of
its symbol.
Reviewer: Vladimir V. Kisil (Leeds)A more accurate multidimensional Hardy-Hilbert's inequality.https://zbmath.org/1453.260352021-02-27T13:50:00+00:00"Yang, Bicheng"https://zbmath.org/authors/?q=ai:yang.bichengSummary: In this paper, by the use of the weight coefficients, the transfer formula, Hermite-Hadamard's inequality and the technique of real analysis, a more accurate multidimensional Hardy-Hilbert's inequality with multi-parameters and a best possible constant factor is given, which is an extension of some published results. Moreover, the equivalent forms and the operator expressions are considered.Positive Gorenstein ideals.https://zbmath.org/1453.141292021-02-27T13:50:00+00:00"Blekherman, Grigoriy"https://zbmath.org/authors/?q=ai:blekherman.grigoriySummary: We introduce positive Gorenstein ideals. These are Gorenstein ideals in the graded ring \( \mathbb{R}[x]\) with socle in degree \( 2d\) which when viewed as a linear functional on \( \mathbb{R}[x]_{2d}\) is nonnegative on squares. Equivalently, positive Gorenstein ideals are apolar ideals of forms whose differential operator is nonnegative on squares. Positive Gorenstein ideals arise naturally in the context of nonnegative polynomials and sums of squares, and they provide a powerful framework for studying concrete aspects of sums of squares representations. We present applications of positive Gorenstein ideals in real algebraic geometry, analysis and optimization. In particular, we present a simple proof of Hilbert's nearly forgotten result on representations of ternary nonnegative forms as sums of squares of rational functions. Drawing on our previous work [J. Am. Math. Soc. 25, No. 3, 617--635 (2012; Zbl 1258.14067)], our main tools are Cayley-Bacharach duality and elementary convex geometry.Approximation of lower bound for matrix operators on the weighted sequence space \(C_p^r(w)(p \in (0,1))\).https://zbmath.org/1453.260332021-02-27T13:50:00+00:00"Talebi, Gholamreza"https://zbmath.org/authors/?q=ai:talebi.gholamreza"Salmei, Hossein"https://zbmath.org/authors/?q=ai:salmei.hosseinSummary: Let \(A=(a_{n,k})_{n,k\geq 0}\) be a non-negative matrix. We denote by \(L_{\ell_p(w),C_{q}^r(w)}(A)\) the supremum of those \(\ell,\) satisfying the following inequality:
\[
\left(\sum_{n = 0}^\infty w_n\left( \frac{1}{(1+r)^n}\sum_{k=n}^\infty \frac{(1+r)^k}{1+k}\sum_{j = 0}^\infty a_{k,j}x_j \right)^q \right)^{1/q} \geq \ell \left(\sum_{n = 0}^\infty w_nx_n^p \right)^{1/p},
\]
where \(x\geq 0\), \(x\in \ell_p(w)\), \(r\in (0,1)\), \(q\leq p\) are numbers in \((0,1)\) and \(\left(w_n \right)_{n=0}^\infty\) is a non-negative and non-increasing sequence of real numbers. In this paper, first we introduce the weighted sequence space \(C_{p}^r(w)~(p \in (0,1))\) of non-absolute type which is a \(p\)-normed space and is isometrically isomorphic to the space \(\ell_p(w)\). Then we focus on the evaluation of \(L_{\ell_p(w),C_{q}^r(w)}(A^t)\) for a lower triangular matrix \(A\), where \(q\leq p\) are in \((0,1)\). A lower estimate is obtained. Moreover, in this paper a Hardy type formula is obtained for \(L_{\ell_p,C_{q}^r}(H_\mu ^\alpha )\) where \(H^\alpha_\mu\) is the generalized Hausdorff matrix, \(q\leq p\leq1\) are positive numbers and \(\alpha\geq 0.\) A similar result is also established for the case in which \(H^\alpha_\mu\) is replaced by \({(H_\mu ^\alpha )^t}\).Numerical radius inequalities for certain \(2\times 2\) operator matrices. II.https://zbmath.org/1453.470012021-02-27T13:50:00+00:00"Shebrawi, Khalid"https://zbmath.org/authors/?q=ai:shebrawi.khalidSummary: We give several sharp numerical radius inequalities for certain \(2 \times 2\) operator matrices. Among other inequalities, it is shown that, if \(A\) and \(B\) be operators in \(\mathfrak{B}(\mathcal{H})\). Then
\[
w\left(
\begin{bmatrix}
A & B \\
0 & 0
\end{bmatrix}
\right) \leq \frac{1}{2}(\| A \| + \| A A^\ast + B B^\ast \|^{\frac{1}{2}})
\]
and
\[
\begin{aligned}
2 w\left(
\begin{bmatrix}
0 & A \\
B & 0
\end{bmatrix}
\right) \leq \max(\| A \|, \| B \|) + \frac{1}{2}(\left\| | A |^t | B^\ast |^{1 - t} \right\| + \left\| | B |^t | A^\ast |^{1 - t} \right\|)
\end{aligned}
\]
for all \(t \in [0, 1]\), where \(w(\cdot)\) and \(\| \cdot \|\) denote the numerical radius and the usual operator norm, respectively. The second inequality refines and generalizes earlier inequalities.
For Part I, see [\textit{O. Hirzallah} et al., Integral Equations Oper. Theory 71, No. 1, 129--147 (2011; Zbl 1238.47004)].Ranks of operators in simple \(C^*\)-algebras with stable rank one.https://zbmath.org/1453.460552021-02-27T13:50:00+00:00"Thiel, Hannes"https://zbmath.org/authors/?q=ai:thiel.hannesIt is obtained that the TW (Toms-Winter) conjecture holds for AsH (approximately subhomogeneous) \(C^*\)-algebras with SR (stable rank) 1. In particular, it is obtained that the JS (Jiang-Su algebra) stability and the SC (strict comparison) property of positive elements are equivalent for SU (simple, separable, unital, non-elementary) \(C^*\)-algebras with SR 1 and ND (nuclear dimension) locally finite. It is shown that for such \(C^*\)-algebras \(A\), any PCA (strictly positive, lower semi-continuous, affine) function on the simplex of normalized quasi-traces of \(A\) is equal to the PCA rank function for some positive element of \(A\) tensored with the \(C^*\)-algebra \(\mathbb K\) of compact operators. Historical motivation for considering comparison and rank theory is given in the lengthy Introduction.
Reviewer: Takahiro Sudo (Nishihara)Collocation of next-generation operators for computing the basic reproduction number of structured populations.https://zbmath.org/1453.920062021-02-27T13:50:00+00:00"Breda, Dimitri"https://zbmath.org/authors/?q=ai:breda.dimitri"Kuniya, Toshikazu"https://zbmath.org/authors/?q=ai:kuniya.toshikazu"Ripoll, Jordi"https://zbmath.org/authors/?q=ai:ripoll.jordi"Vermiglio, Rossana"https://zbmath.org/authors/?q=ai:vermiglio.rossanaSummary: We contribute a full analysis of theoretical and numerical aspects of the collocation approach recently proposed by some of the authors to compute the basic reproduction number of structured population dynamics as spectral radius of certain infinite-dimensional operators. On the one hand, we prove under mild regularity assumptions on the models coefficients that the concerned operators are compact, so that the problem can be properly recast as an eigenvalue problem thus allowing for numerical discretization. On the other hand, we prove through detailed and rigorous error and convergence analyses that the method performs the expected spectral accuracy. Several numerical tests validate the proposed analysis by highlighting diverse peculiarities of the investigated approach.Global multiplicity bounds and spectral statistics for random operators.https://zbmath.org/1453.810272021-02-27T13:50:00+00:00"Mallick, Anish"https://zbmath.org/authors/?q=ai:mallick.anish"Maddaly, Krishna"https://zbmath.org/authors/?q=ai:maddaly.krishnaStrong coupling asymptotics for \(\delta \)-interactions supported by curves with cusps.https://zbmath.org/1453.350592021-02-27T13:50:00+00:00"Flamencourt, Brice"https://zbmath.org/authors/?q=ai:flamencourt.brice"Pankrashkin, Konstantin"https://zbmath.org/authors/?q=ai:pankrashkin.konstantinThe authors consider a simple closed curve \(\Gamma \subset \mathbb{R}^{2}\) which is smooth except at the origin at which it coincides with the curve \(\left\vert x_{2}\right\vert =x_{1}^{p}\), for some \(p>1\), and the Schrödinger operator \(H_{\alpha }\) defined for \(u\in H^{1}(\mathbb{R}^{2})\) as \(H_{\alpha}(u)=\iint\limits_{\mathbb{R}^{2}}\left\vert \nabla u\right\vert^{2}dx-\alpha \int_{\Gamma }u^{2}ds\). The main result of the paper proves that for every \(n\), the eigenvalue \(E_{n}(H_{\alpha })\) is equal to \(-\alpha^{2}+2^{\frac{2}{p+2}}E_{n}(A)\alpha ^{\frac{6}{p+2}}+\mathcal{O}(\alpha ^{\frac{6}{p+2}-\eta })\), as \(\alpha \) tends to \(+\infty \). Here \(E_{n}(A)\) is the \(n\)-th eigenvalue of the auxiliary one-dimensional operator \(A\) acting in \(L^{2}(0,+\infty )\) as \((Af)(x)=-f^{\prime \prime }(x)+x^{p}f(x)\) and \(\eta =\min \{\frac{p-1}{2(p+2)},\frac{2(p-1)}{(p+1)(p+2)}\}\). For the proof, the authors use the min-max principle for the eigenvalues of self-adjoint operators and they reduce the problem to one in a moving half-plane. They indeed introduce \(\Gamma _{\varepsilon }=\{(x_{1},x_{2}):x_{1}\in (0,\varepsilon )\), \(\left\vert x_{2}\right\vert =x_{1}^{p}\}\) and the operator \(H_{\alpha ,\varepsilon }(u)=\iint\limits_{\mathbb{R}
^{2}}\left\vert \nabla u\right\vert ^{2}dx-\alpha \int_{\Gamma _{\varepsilon}}u^{2}ds\). They evaluate the Rayleigh ratio of \(H_{\alpha ,\varepsilon }\) in terms of that of a self-adjoint operator \(F_{h,b}\) and that of \(H_{\alpha}\) in terms of that of \(F_{h,\varepsilon h^{\frac{1}{1-p}}}\) which leads to the introduction of one-dimensional effective operator.
Reviewer: Alain Brillard (Riedisheim)Inverse problems. Basics, theory and applications in geophysics. 2nd updated edition.https://zbmath.org/1453.650022021-02-27T13:50:00+00:00"Richter, Mathias"https://zbmath.org/authors/?q=ai:richter.mathiasPublisher's description: This textbook is an introduction to the subject of inverse problems with an emphasis on practical solution methods and applications from geophysics. The treatment is mathematically rigorous, relying on calculus and linear algebra only; familiarity with more advanced mathematical theories like functional analysis is not required. Containing up-to-date methods, this book will provide readers with the tools necessary to compute regularized solutions of inverse problems. A variety of practical examples from geophysics are used to motivate the presentation of abstract mathematical ideas, thus assuring an accessible approach.Beginning with four examples of inverse problems, the opening chapter establishes core concepts, such as formalizing these problems as equations in vector spaces and addressing the key issue of ill-posedness. Chapter Two then moves on to the discretization of inverse problems, which is a prerequisite for solving them on computers. Readers will be well-prepared for the final chapters that present regularized solutions of inverse problems in finite-dimensional spaces, with Chapter Three covering linear problems and Chapter Four studying nonlinear problems. Model problems reflecting scenarios of practical interest in the geosciences, such as inverse gravimetry and full waveform inversion, are fully worked out throughout the book. They are used as test cases to illustrate all single steps of solving inverse problems, up to numerical computations. Five appendices include the mathematical foundations needed to fully understand the material.This second edition expands upon the first, particularly regarding its up-to-date treatment of nonlinear problems. Following the author's approach, readers will understand the relevant theory and methodology needed to pursue more complex applications. \textit{Inverse Problems} is ideal for graduate students and researchers interested in geophysics and geosciences.
See the review of the first edition in [Zbl 1365.65157].Extraction and prediction of coherent patterns in incompressible flows through space-time koopman analysis.https://zbmath.org/1453.761792021-02-27T13:50:00+00:00"Giannakis, Dimitrios"https://zbmath.org/authors/?q=ai:giannakis.dimitrios"Das, Suddhasattwa"https://zbmath.org/authors/?q=ai:das.suddhasattwaSummary: We develop methods for detecting and predicting the evolution of coherent spatiotemporal patterns in incompressible, time-dependent fluid flows driven by ergodic dynamical systems. Our approach is based on representations of the generators of the Koopman and Perron-Frobenius groups of operators governing the evolution of observables and probability measures on Lagrangian tracers, respectively, in a smooth orthonormal basis learned from velocity field snapshots through the diffusion maps algorithm. These operators are defined on the product space between the state space of the fluid flow and the spatial domain in which the flow takes place, and as a result their eigenfunctions correspond to global space-time coherent patterns under a skew-product dynamical system. Moreover, using this data-driven representation of the generators in conjunction with Leja interpolation for matrix exponentiation, we construct model-free prediction schemes for the evolution of observables and probability densities defined on the tracers. We present applications to periodic Gaussian vortex flows and aperiodic flows generated by Lorenz 96 systems.Representation of bounded solutions of linear discrete equations.https://zbmath.org/1453.390092021-02-27T13:50:00+00:00"Slyusarchuk, V. Yu."https://zbmath.org/authors/?q=ai:slyusarchuk.vasyl-yuGiven a countable set \(G\) and a collection of finite-dimensional Banach spaces \(E_{g}\) (\(g\in G\)), denote by \(\mathcal{M}\) the set of mappings \(x:G\to \bigcup_{g\in G}E_{g}\) such that \(x(g)\in E_{g}\) for all \(g\in G\) and \(\|x\|_{\mathcal{M}}=\sup_{g\in G}\|x(g)\|_{E_{g}}<\infty\). Consider the equation
\[
\sum_{\alpha\in G}A(g,\alpha)x(\alpha)=y(g)\in\mathcal{M},\ g\in G,\tag{1}
\]
where \(A(g,\alpha)\in L(E_{\alpha},E_{g})\). The main result is that, if equation (1) is uniquely solvable for every \(y(g)\in\mathcal{M}\) then there exist linear operators \(\Omega(g,\alpha)\in L(E_{\alpha},E_{g})\) such that a solution \(x(g)\in\mathcal{M}\) to (1) is representable as
\[
x(g)=\sum_{\alpha\in G}\Omega(g,\alpha)y(\alpha),\ \ \sup_{g\in G}\sum_{\alpha\in G}\|\Omega(g,\alpha)\|_{L(E_{\alpha},E_{g})}<\infty.
\]
Reviewer: Sergey G. Pyatkov (Khanty-Mansiysk)Core inverse in Banach algebras.https://zbmath.org/1453.460452021-02-27T13:50:00+00:00"Mosić, Dijana"https://zbmath.org/authors/?q=ai:mosic.dijana"Li, Tingting"https://zbmath.org/authors/?q=ai:li.tingting"Chen, Jianlong"https://zbmath.org/authors/?q=ai:chen.jianlongSummary: We define and characterize the core inverse in the context of Banach algebras. The Banach space operator case is also considered. Using the core inverse, we present new characterizations of EP Banach space operators and EP Banach algebra elements. The dual core inverse for Banach algebra elements is presented too. Some new characterizations of co-EP Banach algebra elements are given by means the core inverse and dual core inverse.Spectral analysis of the spin-boson Hamiltonian with two bosons for arbitrary coupling and bounded dispersion relation.https://zbmath.org/1453.810262021-02-27T13:50:00+00:00"Ibrogimov, Orif O."https://zbmath.org/authors/?q=ai:ibrogimov.orif-o\( \mu \)-norm of an operator.https://zbmath.org/1453.810202021-02-27T13:50:00+00:00"Treschev, D. V."https://zbmath.org/authors/?q=ai:treshchev.dmitrij-vSummary: Let \((\mathcal{X},\mu)\) be a measure space. For any measurable set \(Y\subset\mathcal{X}\) let \(1_Y: \mathcal{X}\to\mathbb{R}\) be the indicator of \(Y\) and let \(\pi_Y\) be the orthogonal projection \(L^2(\mathcal{X})\ni f\mapsto \pi_Y f = 1_Y f\). For any bounded operator \(W\) on \(L^2(\mathcal{X},\mu)\) we define its \(\mu \)-norm \(\|W\|_\mu = \inf_\chi\sqrt{\sum\mu(Y_j)\|W\pi_Y\|^2}\), where the infimum is taken over all measurable partitions \(\chi=\{Y_1,\dots,Y_J\}\) of \(\mathcal{X}\). We present some properties of the \(\mu \)-norm and some computations. Our main motivation is the problem of constructing a quantum entropy.A new index theorem for monomial ideals by resolutions.https://zbmath.org/1453.190072021-02-27T13:50:00+00:00"Douglas, Ronald G."https://zbmath.org/authors/?q=ai:douglas.ronald-george"Jabbari, Mohammad"https://zbmath.org/authors/?q=ai:jabbari.mohammad"Tang, Xiang"https://zbmath.org/authors/?q=ai:tang.xiang"Yu, Guoliang"https://zbmath.org/authors/?q=ai:yu.guoliangSummary: We prove an index theorem for the quotient module of a monomial ideal. We obtain this result by resolving the monomial ideal by a sequence of essentially normal Hilbert modules, each of which is a direct sum of (weighted) Bergman spaces on balls.The base change in the Atiyah and the Lück approximation conjectures.https://zbmath.org/1453.200072021-02-27T13:50:00+00:00"Jaikin-Zapirain, Andrei"https://zbmath.org/authors/?q=ai:jaikin-zapirain.andreiSummary: Let \(F\) be a free finitely generated group and \({A \in \mathrm{Mat}_{n \times m}(\mathbb{C}[F])}\). For each quotient \(G = F/N\) of \(F\) we can define a von Neumann rank function \(\mathrm{rk}_{G}(A)\) associated with the \(l^2\)-operator \(l^2(G)^n \rightarrow l^2(G)^m\) induced by right multiplication by \(A\). For example, in the case where \(G\) is finite, \(\mathrm{rk}_G(A)=\frac{\mathrm{rk}_{\mathbb{C}}(\bar{A})}{|G|}\) is the normalized rank of the matrix \(\bar{A} \in \mathrm{Mat}_{n \times m}(\mathbb{C}[G])\) obtained by reducing the coefficients of \(A\) modulo \(N\). One of the variations of the Lück approximation conjecture claims that the function \({N\mapsto \mathrm{rk}_{F/N}(A)}\) is continuous in the space of marked groups. The strong Atiyah conjecture predicts that if the least common multiple lcm(G) of the orders of finite subgroups of \(G\) is finite, then \({\mathrm{rk}_G(A) \in \frac{1}{\mathrm{lcm} (G)}\mathbb{Z}}\). In our first result we prove the sofic Lück approximation conjecture. In particular, we show that the function \({N \mapsto \mathrm{rk}_{F/N}(A)}\) is continuous in the space of sofic marked groups. Among other consequences we obtain that a strong version of the algebraic eigenvalue conjecture, the center conjecture and the independence conjecture hold for sofic groups. In our second result we apply the sofic Lück approximation and we show that the strong Atiyah conjecture holds for groups from a class \({{\mathcal{D}}}\), virtually compact special groups, Artin's braid groups and torsion-free \(p\)-adic analytic pro-\(p\) groups.