Recent zbMATH articles in MSC 47Ahttps://zbmath.org/atom/cc/47A2022-11-17T18:59:28.764376ZWerkzeugStationary measures on infinite graphshttps://zbmath.org/1496.051162022-11-17T18:59:28.764376Z"Baraviera, Alexandre"https://zbmath.org/authors/?q=ai:baraviera.alexandre-tavares"Duarte, Pedro"https://zbmath.org/authors/?q=ai:duarte.pedro"Torres, Maria Joana"https://zbmath.org/authors/?q=ai:torres.maria-joanaTwo arithmetic applications of perturbations of composition operatorshttps://zbmath.org/1496.110382022-11-17T18:59:28.764376Z"Bettin, Sandro"https://zbmath.org/authors/?q=ai:bettin.sandro"Drappeau, Sary"https://zbmath.org/authors/?q=ai:drappeau.saryAuthors' abstract: We estimate the spectral radius of perturbations of a particular family of composition operators, in a setting where the usual choices of norms do not account for the typical size of the perturbation. We apply this to estimate the growth rate of large moments of a Thue-Morse generating function and of the Stern sequence. This answers in particular a question of \textit{C. Mauduit} et al. [J. Anal. Math. 135, No. 2, 713--724 (2018; Zbl 1448.11059)].
Reviewer: Michel Rigo (Liège)Towards a fractal cohomology: spectra of Polya-Hilbert operators, regularized determinants and Riemann zeroshttps://zbmath.org/1496.111202022-11-17T18:59:28.764376Z"Cobler, Tim"https://zbmath.org/authors/?q=ai:cobler.tim"Lapidus, Michel L."https://zbmath.org/authors/?q=ai:lapidus.michel-lSummary: Emil Artin defined a zeta function for algebraic curves over finite fields and made a conjecture about them analogous to the famous Riemann hypothesis. This and other conjectures about these zeta functions would come to be called the Weil conjectures, which were proved by Weil in the case of curves and eventually, by Deligne in the case of varieties over finite fields. Much work was done in the search for a proof of these conjectures, including the development in algebraic geometry of a Weil cohomology theory for these varieties, which uses the Frobenius operator on a finite field. The zeta function is then expressed as a determinant, allowing the properties of the function to relate to the properties of the operator. The search for a suitable cohomology theory and associated operator to prove the Riemann hypothesis has continued to this day. In this paper we study the properties of the derivative operator \(D = \frac{d} {dz}\) on a particular family of weighted Bergman spaces of entire functions on \(\mathbb{C}\). The operator \(D\) can be naturally viewed as the ``infinitesimal shift of the complex plane'' since it generates the group of translations of \(\mathbb{C}\). Furthermore, this operator is meant to be the replacement for the Frobenius operator in the general case and is used to construct an operator associated with any given meromorphic function. With this construction, we show that for a wide class of meromorphic functions, the function can be recovered by using a regularized determinant involving the operator constructed from the meromorphic function. This is illustrated in some important special cases: rational functions, zeta functions of algebraic curves (or, more generally, varieties) over finite fields, the Riemann zeta function, and culminating in a quantized version of the Hadamard factorization theorem that applies to any entire function of finite order. This shows that all of the information about the given meromorphic function is encoded into the special operator we constructed. Our construction is motivated in part by work of Herichi and the second author on the infinitesimal shift of the real line (instead of the complex plane) and the associated spectral operator, as well as by earlier work and conjectures of Deninger on the role of cohomology in analytic number theory, and a conjectural ``fractal cohomology theory'' envisioned in work of the second author and of Lapidus and van Frankenhuijsen on complex fractal dimensions.
For the entire collection see [Zbl 1381.11005].Logarithmic mean of multiple accretive matriceshttps://zbmath.org/1496.150152022-11-17T18:59:28.764376Z"Luo, Wenhui"https://zbmath.org/authors/?q=ai:luo.wenhuiSummary: Using the integral representation of logarithmic mean, we define the logarithmic mean of multiple accretive matrices. When the number of matrices is two, it coincides with the recent studies carried out by \textit{F. Tan} and \textit{A. Xie} [Filomat 33, No. 15, 4747--4752 (2019; Zbl 07537437)]. Several inequalites are presented along with the studies.Clarke Jacobians, Bouligand Jacobians, and compact connected sets of matriceshttps://zbmath.org/1496.260122022-11-17T18:59:28.764376Z"Bartl, David"https://zbmath.org/authors/?q=ai:bartl.david"Fabian, Marián"https://zbmath.org/authors/?q=ai:fabian.marian-j"Kolář, Jan"https://zbmath.org/authors/?q=ai:kolar.janThis note is dedicated to extending from Clarke Jacobians to Bouligand Jacobians various recent results of the first two named authors. The main statement reveals that every nonempty compact connected set of matrices can be expressed as the Bouligand Jacobian at the origin of a suitable Lipschitzian mapping which is moreover either countably piecewise affine or \(C^\infty\)-smooth outside the neighbourhoods of the origin.
Reviewer: Sorin-Mihai Grad (Paris)The abstract Cauchy problem in locally convex spaceshttps://zbmath.org/1496.340282022-11-17T18:59:28.764376Z"Kruse, Karsten"https://zbmath.org/authors/?q=ai:kruse.karstenSummary: We derive necessary and sufficient criteria for the uniqueness and existence of solutions of the abstract Cauchy problem in locally convex Hausdorff spaces. Our approach is based on a suitable notion of an asymptotic Laplace transform and extends results of Langenbruch beyond the class of Fréchet spaces.Complete asymptotic expansions of the spectral function for symbolic perturbations of almost periodic Schrödinger operators in dimension onehttps://zbmath.org/1496.341272022-11-17T18:59:28.764376Z"Galkowski, Jeffrey"https://zbmath.org/authors/?q=ai:galkowski.jeffreyThe abstract of the paper itself includes the most accurate and complete explanation about the content of the article under review. Here we quote it in full:
``In this article we consider asymptotics for the spectral function of Schrödinger operators on the real line. Let \(P\colon L^2(\mathbb{R})\to L^2(\mathbb{R})\) have the form \(P:=-\frac{d^2}{dx^2}+W\), where \(W\) is a self-adjoint first order differential operator with certain modified almost periodic structure. We show that the kernel of the spectral projector, \(\mathbf{1}_{(-\infty,\lambda^2]}(P)\) has a full asymptotic expansion in powers of \(\lambda\). In particular, our class of potentials \(W\) is stable under perturbation by formally self-adjoint first order differential operators with smooth, compactly supported coefficients. Moreover, the class of potentials includes certain potentials with dense pure point spectrum. The proof combines the gauge transform methods of Parnovski-Shterenberg and Sobolev with Melrose's scattering calculus.''
Reviewer: Erdogan Sen (Tekirdağ)Scattering theory for transport phenomena. With a foreword by Peter Laxhttps://zbmath.org/1496.350022022-11-17T18:59:28.764376Z"Emamirad, Hassan"https://zbmath.org/authors/?q=ai:emamirad.hassan-aliIn this book, the author develops a part of the progress of the scattering theory for transport phenomena. Scattering theory is a powerful technical tool in mathematical physics, and transport theory falls within the province of statistical physics.
This book is divided into seven chapters. In Chapter 1, as a preliminaries, the author gives the theory of semigroups and C*-algebra, different types of semigroups, Schatten-von Neumann classes of operators, and some facts about ultraweak operator topology.
In Chapter 2, the author goes into the abstract scattering theory in a general Banach space, and defines the wave and scattering operators and their basic properties. Some abstract methods such as smooth perturbations and the limiting absorption principle are presented.
Chapter 3 is devoted to the transport or linearized Boltzmann equation, which is the advection equation perturbed by the sum of absorption and production operators.
In Chapter 4, the author introduces the Lax and Phillips formalism in scattering theory for the transport equation.
Chapter 5 is devoted to introduce the scattering theory for a charged particle transport problem.
Chapter 6 is the highlight of the book in which the author explains how the scattering operator for the transport problem can lead us to formulate the computerized tomography in medical science.
In the last chapter, the author introduces the Wigner function and shows how this function connects the Schrödinger equation to statistical physics and the Husimi distribution function.
Reviewer: Jiqiang Zheng (Beijing)Generation of analytic semigroups for some generalized diffusion operators in \(L^p\)-spaceshttps://zbmath.org/1496.351332022-11-17T18:59:28.764376Z"Labbas, Rabah"https://zbmath.org/authors/?q=ai:labbas.rabah"Maingot, Stéphane"https://zbmath.org/authors/?q=ai:maingot.stephane"Thorel, Alexandre"https://zbmath.org/authors/?q=ai:thorel.alexandreSummary: We consider some generalized diffusion operators of fourth order and their corresponding abstract Cauchy problem. Then, using semigroups techniques and functional calculus, we study the invertibility and the spectral properties of each operator. Therefore, we prove that we have generation of \(C_0\)-semigroup in each case. We also point out when these semigroups become analytic.Uniform \(L^p\) resolvent estimates on the torushttps://zbmath.org/1496.351732022-11-17T18:59:28.764376Z"Hickman, Jonathan"https://zbmath.org/authors/?q=ai:hickman.jonathanThe author presents a new range of uniform \(L^p\) resolvent estimates, in the setting of the flat torus \({\mathbb T}^N:={\mathbb R}^N/{\mathbb Z}^N\) for \(N\ge 3\), improving previous results. Specifically, if \(\Delta_{{\mathbb T}^N}\) represents the Laplacian on the flat torus, then for all \(\varepsilon>0\) there exists a \(C_\varepsilon>0\) such that for all
\[
z\in\{z=(\lambda + i\mu)^2\in\mathbb{C}:\, \lambda, \mu\in\mathbb{R},\ \lambda\ge 1,\, |\mu|\ge \lambda^{-1/3+\varepsilon}\},
\]
the following holds
\[
\|u\|_{L^{2^*}({\mathbb T}^N)}\le C_\varepsilon\, \big\|(\Delta_{{\mathbb T}^N}+z)u\big\|_{L^{(2^*)'}({\mathbb T}^N)},
\]
where \(2^*:=\frac{2N}{N-2}\) is the critical Sobolev exponent, and \((2^*)'=\frac{2N}{N+2}\) is its conjugate.
Their arguments rely on the \(l^2\)-decoupling theorem and multidimensional Weyl sum estimates.
Reviewer: Rosa Maria Pardo San Gil (Madrid)On the limiting absorption principle for a new class of Schrödinger Hamiltonianshttps://zbmath.org/1496.351802022-11-17T18:59:28.764376Z"Martin, Alexandre"https://zbmath.org/authors/?q=ai:martin.alexandre.1|martin.alexandreSummary: We prove the limiting absorption principle and discuss the continuity properties of the boundary values of the resolvent for a class of form bounded perturbations of the Euclidean Laplacian \(\Delta\) that covers both short and long range potentials with an essentially optimal behaviour at infinity. For this, we give an extension of \textit{S. Nakamura}'s results [J. Spectr. Theory 4, No. 3, 613--619 (2014; Zbl 1308.81165)].Principal spectral theory and asynchronous exponential growth for age-structured models with nonlocal diffusion of Neumann typehttps://zbmath.org/1496.352592022-11-17T18:59:28.764376Z"Kang, Hao"https://zbmath.org/authors/?q=ai:kang.hao"Ruan, Shigui"https://zbmath.org/authors/?q=ai:ruan.shiguiSummary: In this paper we study the principal spectral theory and asynchronous exponential growth for age-structured models with nonlocal diffusion of Neumann type. First, we provide two general sufficient conditions to guarantee existence of the principal eigenvalue of the age-structured operator with nonlocal diffusion. Then we show that such conditions are also enough to ensure that the semigroup generated by solutions of the age-structured model with nonlocal diffusion exhibits asynchronous exponential growth. Compared with previous studies, we prove that the semigroup is essentially compact instead of eventually compact, where the latter is usually obtained by showing the compactness of solution trajectories. Next, following the technique developed in Vo (Principal spectral theory of time-periodic nonlocal dispersal operators of Neumann type. arXiv:1911.06119, 2019), we overcome the difficulty that the principal eigenvalue of a nonlocal Neumann operator is not monotone with respect to the domain and obtain some limit properties of the principal eigenvalue with respect to the diffusion rate and diffusion range. Finally, we establish the strong maximum principle for the age-structured operator with nonlocal diffusion.Bogoliubov theory for many-body quantum systemshttps://zbmath.org/1496.353352022-11-17T18:59:28.764376Z"Schlein, Benjamin"https://zbmath.org/authors/?q=ai:schlein.benjaminSummary: We review some recent applications of rigorous Bogoliubov theory. We show how Bogoliubov theory can be used to approximate quantum fluctuations, both in the analysis of the energy spectrum and in the study of the dynamics of many-body quantum systems.
For the entire collection see [Zbl 1465.35005].Global subelliptic estimates for Kramers-Fokker-Planck operators with some class of polynomialshttps://zbmath.org/1496.353872022-11-17T18:59:28.764376Z"Ben Said, Mona"https://zbmath.org/authors/?q=ai:said.mona-benSummary: In this article, we study some Kramers-Fokker-Planck operators with a polynomial potential \(V(q)\) of degree greater than two having quadratic limiting behaviour. This work provides an accurate global subelliptic estimate for Kramers-Fokker-Planck operators under some conditions imposed on the potential.On essential numerical ranges and essential spectra of Hamiltonian systems with one singular endpointhttps://zbmath.org/1496.370562022-11-17T18:59:28.764376Z"Zhu, Li"https://zbmath.org/authors/?q=ai:zhu.li"Sun, Huaqing"https://zbmath.org/authors/?q=ai:sun.huaqingSummary: This paper is concerned with essential numerical ranges and essential spectra of Hamiltonian systems with one singular endpoint. For semi-bounded systems, the characterization of each element of the essential numerical range in terms of certain singular sequences is given, the concept of form perturbation small at the singular endpoint is introduced, and the stability of the essential numerical range is obtained under this perturbation, which shows the stability of the infimum or supremum of the essential spectrum. Some sufficient conditions for the invariance of the essential numerical range are given in terms of coefficients of Hamiltonian systems.On the spectral and scattering properties of eigenparameter dependent discrete impulsive Sturm-Liouville equationshttps://zbmath.org/1496.390112022-11-17T18:59:28.764376Z"Aygar Küçükevcilioğlu, Yelda"https://zbmath.org/authors/?q=ai:aygar-kucukevcilioglu.yelda"Bayram, Elgiz"https://zbmath.org/authors/?q=ai:bayram.elgiz"Özbey, Güher Gülçehre"https://zbmath.org/authors/?q=ai:ozbey.guher-gulcehreSummary: This work develops scattering and spectral analysis of a discrete impulsive Sturm-Liouville equation with spectral parameter in boundary condition. Giving the Jost solution and scattering solutions of this problem, we find scattering function of the problem. Discussing the properties of scattering function, scattering solutions, and asymptotic behavior of the Jost solution, we find the Green function, resolvent operator, continuous and point spectrum of the problem. Finally, we give an example in which the main results are made explicit.Positive definiteness and infinite divisibility of certain functions of hyperbolic cosine functionhttps://zbmath.org/1496.420102022-11-17T18:59:28.764376Z"Kosaki, Hideki"https://zbmath.org/authors/?q=ai:kosaki.hidekiLet \(\alpha \geq 0\), \(t>-1\) and \(f_{\alpha },\) \(g_{\alpha }\) two real functions defined by
\[
f_{\alpha }(x)=\frac{\cosh (\alpha x)}{\cosh (3x)+t\cosh x}
\]
and
\[
g_{\alpha }(x)=\frac{\cosh (\alpha x)}{\cosh (2x)+t\cosh x}.
\]
The author investigates infinite divisibility and positive definiteness of the functions \(f_{\alpha }\) and of \(g_{\alpha }\). Furthermore, he uses the positive definiteness criterion to study certain norm comparison results for operator means.
Reviewer: Elhadj Dahia (Bou Saâda)Order structures of \((\mathcal{D,E})\)-quasi-bases and constructing operators for generalized Riesz systemshttps://zbmath.org/1496.420432022-11-17T18:59:28.764376Z"Inoue, Hiroshi"https://zbmath.org/authors/?q=ai:inoue.hiroshiSummary: The main purpose of this paper is to investigate the relationship between the two order structures of constructing operators for a generalized Riesz system and \((\mathcal{D,E})\)-quasi bases for two fixed biorthogonal sequences \(\{\varphi_n\}\) and \(\{\Psi_n\}\). In a previous paper, we have studied the order structure of the set \(C_\varphi\) of all constructing operators for a generalized Riesz system \(\{\varphi_n\}\), and furthermore we have shown that the notion of generalized Riesz systems has a close relation with that of \((\mathcal{D,E})\)-quasi bases. For this reason, in this paper we define an order structure in the set \(\mathfrak{M}_{\varphi,\psi}\) of all pairs of dense subspaces \(\mathcal{D}\) and \(\mathcal{E}\) in \(\mathcal{H}\) such that \(\{\varphi_n\}\) and \(\{\psi_n\}\) are \((\mathcal{D,E})\)-quasi bases, and shall investigate the relationships between the order sets \(C_\varphi\), \(C_\psi\) and \(M_{\varphi,\psi}\). These results seem to be useful to find suitable constructing operators for each physical model.Almost multiplicative maps and \(\varepsilon\)-spectrum of an element in Fréchet \(Q\)-algebrahttps://zbmath.org/1496.460452022-11-17T18:59:28.764376Z"Farajzadeh, A. P."https://zbmath.org/authors/?q=ai:farajzadeh.ali-p"Omidi, M. R."https://zbmath.org/authors/?q=ai:omidi.mohammad-rezaSummary: Let \((A,(p_k))\) be a Fréchet \(Q\)-algebra with unit \(e_A\). The \(\varepsilon\)-spectrum of an element \(x\) in \(A\) is defined by
\[
\sigma_\varepsilon(x)=\left\{\lambda\in\mathbb{C}:p_{k_0}(\lambda e_A-x)p_{k_0}(\lambda e_A-x)^{-1}\geq \frac{1}{\varepsilon}\right\}
\]
for \(0<\varepsilon<1\). We show that there is a close relation between the
\(\varepsilon\)-spectrum and almost multiplicative maps. It is also shown that
\[
\{\varphi(x):\varphi\in M^\varepsilon_{alm}(A),\varphi(e_A)=1\}\subseteq\sigma_\varepsilon(x)
\]
for every \(x\in A\), where \(M^\varepsilon_{alm}(A)\) is the set of all \(\varepsilon\)-multiplicative maps from \(A\) to \(\mathbb{C}\).Noncommutative Choquet simpliceshttps://zbmath.org/1496.460542022-11-17T18:59:28.764376Z"Kennedy, Matthew"https://zbmath.org/authors/?q=ai:kennedy.matthew"Shamovich, Eli"https://zbmath.org/authors/?q=ai:shamovich.eliSummary: We introduce a notion of noncommutative Choquet simplex, or briefly an nc simplex, that generalizes the classical notion of a simplex. While every simplex is an nc simplex, there are many more nc simplices. They arise naturally from \(C^*\)-algebras and in noncommutative dynamics. We characterize nc simplices in terms of their geometry and in terms of structural properties of their corresponding operator systems. There is a natural definition of nc Bauer simplex that generalizes the classical definition of a Bauer simplex. We show that a compact nc convex set is an nc Bauer simplex if and only if it is affinely homeomorphic to the nc state space of a unital \(C^*\)-algebra, generalizing a classical result of Bauer for unital commutative \(C^*\)-algebras. We obtain several applications to noncommutative dynamics. We show that the set of nc states of a \(C^*\)-algebra that are invariant with respect to the action of a discrete group is an nc simplex. From this, we obtain a noncommutative ergodic decomposition theorem with uniqueness. Finally, we establish a new characterization of discrete groups with Kazhdan's property (T) that extends a result of \textit{E. Glasner} and \textit{B. Weiss} [Geom. Funct. Anal. 7, No. 5, 917--935 (1997; Zbl 0899.22006)]. Specifically, we show that a discrete group has property (T) if and only if for every action of the group on a unital \(C^*\)-algebra, the set of invariant states is affinely homeomorphic to the state space of a unital \(C^*\)-algebra.A generalized powers averaging property for commutative crossed productshttps://zbmath.org/1496.460572022-11-17T18:59:28.764376Z"Amrutam, Tattwamasi"https://zbmath.org/authors/?q=ai:amrutam.tattwamasi"Ursu, Dan"https://zbmath.org/authors/?q=ai:ursu.danPowers' averaging property for discrete groups has played an important role in questions about simplicity related to reduced group \(C^\ast\)-algebras and reduced crossed products. In the present paper, the authors introduce a generalized version of Powers' averaging property for reduced crossed products of group \(C^\ast\)-algebras, and prove that it is equivalent to simplicity of the crossed product.
Reviewer: Luoyi Shi (Tianjin)Essentially invertible measurable operators affiliated to a semifinite von Neumann algebra and commutatorshttps://zbmath.org/1496.460622022-11-17T18:59:28.764376Z"Bikchentaev, A. M."https://zbmath.org/authors/?q=ai:bikchentaev.airat-mSummary: Suppose that a von Neumann operator algebra \({\mathcal{M}}\) acts on a Hilbert space \({\mathcal{H}}\) and \(\tau\) is a faithful normal semifinite trace on \({\mathcal{M}} \). If Hermitian operators \(X,Y\in S({\mathcal{M}},\tau)\) are such that \(-X\leq Y\leq X\) and \(Y\) is \(\tau \)-essentially invertible then so is \(X \). Let \(0<p\leq 1 \). If a \(p \)-hyponormal operator \(A\in S({\mathcal{M}},\tau)\) is right \(\tau \)-essentially invertible then \(A\) is \(\tau \)-essentially invertible. If a \(p \)-hyponormal operator \(A\in{\mathcal{B}}({\mathcal{H}})\) is right invertible then \(A\) is invertible in \({\mathcal{B}}({\mathcal{H}}) \). If a hyponormal operator \(A\in S({\mathcal{M}},\tau)\) has a right inverse in \(S({\mathcal{M}},\tau)\) then \(A\) is invertible in \(S({\mathcal{M}},\tau) \). If \(A,T\in{\mathcal{M}}\) and \(\mu_t(A^n)^{\frac{1}{n}}\to 0\) as \(n\to\infty\) for every \(t>0\) then \(AT ( TA )\) has no right (left) \( \tau \)-essential inverse in \(S({\mathcal{M}},\tau) \). Suppose that \({\mathcal{H}}\) is separable and \(\dim{\mathcal{H}}=\infty \). A right (left) essentially invertible operator \(A\in{\mathcal{B}}({\mathcal{H}})\) is a commutator if and only if the right (left) essential inverse of \(A\) is a commutator.On some essential spectra of off-diagonal block operator matrix with applicationhttps://zbmath.org/1496.470022022-11-17T18:59:28.764376Z"He, Ruxia"https://zbmath.org/authors/?q=ai:he.ruxia"Wu, Deyu"https://zbmath.org/authors/?q=ai:wu.deyuSummary: Let \(\mathcal{H}=\begin{pmatrix}0 & B \\ C & 0 \end{pmatrix}\) be an off-diagonal \(2\times 2\) block operator matrix on the product of Hilbert spaces \(X\times X\). In this paper, the essential spectra of \(\mathcal{H}\) is characterized by the essential spectra of \(BC\) and \(CB\). Furthermore, we give an application to infinite dimensional Hamiltonian operator.On the spectral radius of antidiagonal block operator matriceshttps://zbmath.org/1496.470032022-11-17T18:59:28.764376Z"Ipek Al, Pembe"https://zbmath.org/authors/?q=ai:ipek-al.pembe"Ismailov, Zameddin I."https://zbmath.org/authors/?q=ai:ismailov.zameddin-ismailovichSummary: In this paper, the difference between operator norm and spectral radius for the antidiagonal block operator matrix in the direct sum of Hilbert spaces is investigated. Also, the necessary and sufficient conditions for these operators belong to Schatten-von Neumann classes are given.Local spectral property of \(2 \times 2\) operator matriceshttps://zbmath.org/1496.470042022-11-17T18:59:28.764376Z"Ko, Eungil"https://zbmath.org/authors/?q=ai:ko.eungilSummary: In this paper we study the local spectral properties of \(2 \times 2\) operator matrices. In particular, we show that every \(2 \times 2\) operator matrix with three scalar entries has the single valued extension property. Moreover, we consider the spectral properties of such operator matrices. Finally, we show that some of such operator matrices are decomposable.Semigroup generations of unbounded block operator matrices based on the space decompositionhttps://zbmath.org/1496.470052022-11-17T18:59:28.764376Z"Liu, Jie"https://zbmath.org/authors/?q=ai:liu.jie|liu.jie.1|liu.jie.3|liu.jie.4|liu.jie.2|liu.jie.7|liu.jie.5"Huang, Junjie"https://zbmath.org/authors/?q=ai:huang.junjie"Chen, Alatancang"https://zbmath.org/authors/?q=ai:chen.alatancangThe authors find necessary and sufficient conditions under which an unbounded block operator matrix
\[
M = \begin{pmatrix} A & B \\
C & D \end{pmatrix}
\]
with natural domain \(\mathscr{D}(M)=(\mathscr{D}(A)\cap \mathscr{D}(C))\oplus (\mathscr{D}(B)\cap \mathscr{D}(D))\) generates a \(C_0\)-semigroup. Usually, in the literature, most of the results are presented using the diagonal domain \(\mathscr{D}(M) = \mathscr{D}(A)\oplus \mathscr{D}(D)\), using standard perturbation theorems. To prove the results in the natural domain, the authors characterize the right boundedness of \(M\) with the quadratic numerical range of \(M\), and consider the residual spectrum based on the space decomposition and quadratic complements.
Reviewer: Matheus Cheque Bortolan (Florianópolis)The point spectrum and residual spectrum of upper triangular operator matriceshttps://zbmath.org/1496.470062022-11-17T18:59:28.764376Z"Wu, Xiufeng"https://zbmath.org/authors/?q=ai:wu.xiufeng"Huang, Junjie"https://zbmath.org/authors/?q=ai:huang.junjie"Chen, Alatancang"https://zbmath.org/authors/?q=ai:chen.alatancangSummary: The point and residual spectra of an operator are, respectively, split into \(1,2\)-point spectrum and \(1,2\)-residual spectrum, based on the denseness and closedness of its range. Let \(\mathcal{H,K}\) be infinite dimensional complex separable Hilbert spaces and write
\(M_X = \begin{pmatrix} A & X \\ 0 & B \end{pmatrix} \in \mathcal{B}(\mathcal{H} \oplus \mathcal{K})\).
For given operators \(A \in \mathcal{B(H)}\) and \(B \in \mathcal{B(K)}\), the sets \(\bigcup\limits_{X \in \mathcal{B}(\mathcal{K},\mathcal{H})} \sigma_{\ast,i}(M_X) \) (\(* = p,r\); \(i=1,2\)) are characterized.
Moreover, we obtain some necessary and sufficient condition such that \(\sigma_{\ast,i}(M_X) = \sigma_{*,i}(A) \cup \sigma_{*,i}(B)\) (\(* = p,r\); \(i=1,2\)) for every \( X \in \mathcal{B}(\mathcal{K},\mathcal{H})\).On the essential numerical spectrum of operators on Banach spaceshttps://zbmath.org/1496.470072022-11-17T18:59:28.764376Z"Abdelhedi, Bouthaina"https://zbmath.org/authors/?q=ai:abdelhedi.bouthaina"Boubaker, Wissal"https://zbmath.org/authors/?q=ai:boubaker.wissal"Moalla, Nedra"https://zbmath.org/authors/?q=ai:moalla.nedraSummary: The purpose of this paper is to define and develop a new notion of the essential numerical spectrum \(\sigma_{en}(.)\) of an operator on a Banach space \(X\) and to study its properties. Our definition is closely related to the essential numerical range \(W_e(.)\).The spectrum of the restriction to an invariant subspacehttps://zbmath.org/1496.470082022-11-17T18:59:28.764376Z"Drivaliaris, Dimosthenis"https://zbmath.org/authors/?q=ai:drivaliaris.dimosthenis"Yannakakis, Nikos"https://zbmath.org/authors/?q=ai:yannakakis.nikolaosLet \(A\) be a bounded linear operator on a Banach space, \(\rho(A)\) (resp., \(\sigma(A)\)) its resolvent (resp., spectrum). Let \(M\) be a closed invariant subspace of \(A\) and \(D\) be a connected component of \(\rho(A)\). The authors state that, if \(D\cap\sigma(A_{\restriction_M})\neq\emptyset\), then \(D\subset \sigma(A_{\restriction_M})\).
Reviewer: Mohammed El Aïdi (Bogotá)Pseudospectrum enclosures by discretizationhttps://zbmath.org/1496.470092022-11-17T18:59:28.764376Z"Frommer, Andreas"https://zbmath.org/authors/?q=ai:frommer.andreas"Jacob, Birgit"https://zbmath.org/authors/?q=ai:jacob.birgit"Vorberg, Lukas"https://zbmath.org/authors/?q=ai:vorberg.lukas-a"Wyss, Christian"https://zbmath.org/authors/?q=ai:wyss.christian"Zwaan, Ian"https://zbmath.org/authors/?q=ai:zwaan.ian-nLet \(A\) be a matrix (or operator) of finite or infinite dimension. For \(\lambda \in \mathbb{C}\), the resolvent matrix is \(R_\lambda(A)=(A-\lambda)^{-1}\). To understand more about an object represented by a matrix \(A\), we have to analyze not only the eigenvalues, spectrum and resolvent matrix, but also the pseudospectrum. The concept of pseudospectrum of matrices has a number of applications in different fields: dynamical systems, hydrodynamic stability, Markov chains, and non-Hermitian quantum mechanics. Note that the eigenvalues of the resolvent matrix of \(A\) never coincide with the eigenvalues of \(R_\lambda(A)\).
It is well known that for \(\epsilon>0,\) pseudospectrum of \(A\) is given by
\[\sigma_\epsilon(A)=\{\lambda\in \mathbb{C}:\|(A-\lambda)^{-1}\|< \epsilon^{-1}\}.\]
In this paper, a new method is employed to enclose the pseudospectrum via the numerical range of the inverse of the matrix or linear operator.
The results (Theorem 2.2, Theorem 2.5, Theorem 3.6 and Lemma 4.5) will have a significant impact in the area of studying spectral analysis of matrices or linear operators. Also, there are computations given in Example 7.1 and Example 7.2.
Reviewer: Ali Shukur (Minsk)Spectral \(\zeta\)-functions and \(\zeta\)-regularized functional determinants for regular Sturm-Liouville operatorshttps://zbmath.org/1496.470102022-11-17T18:59:28.764376Z"Fucci, Guglielmo"https://zbmath.org/authors/?q=ai:fucci.guglielmo"Gesztesy, Fritz"https://zbmath.org/authors/?q=ai:gesztesy.fritz"Kirsten, Klaus"https://zbmath.org/authors/?q=ai:kirsten.klaus"Stanfill, Jonathan"https://zbmath.org/authors/?q=ai:stanfill.jonathanSummary: The principal aim in this paper is to employ a recently developed unified approach to the computation of traces of resolvents and \(\zeta\)-functions to efficiently compute values of spectral \(\zeta\)-functions at positive integers associated with regular (three-coefficient) self-adjoint Sturm-Liouville differential expressions \(\tau\). Depending on the underlying boundary conditions, we express the \(\zeta\)-function values in terms of a fundamental system of solutions of \(\tau y=zy\) and their expansions about the spectral point \(z=0\). Furthermore, we give the full analytic continuation of the \(\zeta\)-function through a Liouville transformation and provide an explicit expression for the \(\zeta\)-regularized functional determinant in terms of a particular set of this fundamental system of solutions. An array of examples illustrating the applicability of these methods is provided, including regular Schrödinger operators with zero, piecewise constant, and a linear potential on a compact interval.Branching form of the resolvent at thresholds for multi-dimensional discrete Laplacianshttps://zbmath.org/1496.470112022-11-17T18:59:28.764376Z"Ito, Kenichi"https://zbmath.org/authors/?q=ai:ito.kenichi.1"Jensen, Arne"https://zbmath.org/authors/?q=ai:jensen.arneSummary: We consider the discrete Laplacian on \(\mathbb{Z}^d\), and compute asymptotic expansions of its resolvent around thresholds embedded in continuous spectrum as well as those at end points. We prove that the resolvent has a square-root branching if \(d\) is odd, and a logarithm branching if \(d\) is even, and, moreover, obtain explicit expressions for these branching parts involving the Lauricella hypergeometric function. In order to analyze a non-degenerate threshold of general form we use an elementary step-by-step expansion procedure, less dependent on special functions.Hypergeometric expression for the resolvent of the discrete Laplacian in low dimensionshttps://zbmath.org/1496.470122022-11-17T18:59:28.764376Z"Ito, Kenichi"https://zbmath.org/authors/?q=ai:ito.kenichi.2|ito.kenichi.1"Jensen, Arne"https://zbmath.org/authors/?q=ai:jensen.arne-m|jensen.arne-skov|jensen.arneThe authors have obtained some closed formulae for lattice Green functions of the form \[ G(z,n)=(2\pi)^{-d}\int_{\mathbb{T}^d}\dfrac{e^{in\theta}}{2d-2\cos(\theta_1)-\dots-2\cos(\theta_d)-z}d\theta.\]
Such investigation was mainly restricted to dimensions \(d=1,2\). In the Introduction, they start to depict that \(2dG(0,n)\) (\(z=0\)) shall be represented as the expectation value \( \mathbb{E}[n]=\sum_{k=0}^\infty P(X_k=n)\) that counts the number of times that a walker visits \(n\in \mathbb{Z}^d\). To get rid of the fact that \(\mathbb{E}[n]\) is divergent for dimensions \(d=1,2\) (see also Appendix B), they propose a renormalization technique to approximate \(\mathbb{E}[n]\) by \(\mathbb{E}[\epsilon,n]=\frac{2d}{1-\epsilon}G(\frac{-2d\epsilon}{1-\epsilon},n)\), for values of \(\epsilon\in (0,1]\).
In this way, they succeed in representing \(G(z,n)\) as a convergent series (see, e.g., Theorem 2.2. and Theorem 2.3.). Such analysis goes far beyond the asymptotic analysis, in the limit \(z\rightarrow 0\), considered by so many authors in the past.
As a whole, this paper is complementary to the thors' previous paper [J. Funct. Anal. 277, No. 4, 965--993 (2019; Zbl 1496.47011)] in which the authors have shown that \(G(z,n)\) admits, for each threshold \(4q\), \(q=0,\dots,n\), the splitting formula \[ G(z,n)=\mathcal{E}_q(z,n)+f_q(z)\mathcal{F}_q(z,n), \] whereby \(\mathcal{F}_q(z,n)\) -- the singular part of \(G(z,n)\) -- was represented in terms of the so-called Appell-Lauricella hypergeometric function of type \(B\), \(F_B^{(d)}\). Further comparisons between both approaches may be found in Appendix~A.
Reviewer: Nelson Faustino (Alfeizerão)On the numerical range and operator norm of \(V^2\)https://zbmath.org/1496.470132022-11-17T18:59:28.764376Z"Khadkhuu, L."https://zbmath.org/authors/?q=ai:khadkhuu.lkhamzhav|khadkhuu.lkhamjav"Tsedenbayar, D."https://zbmath.org/authors/?q=ai:tsedenbayar.dashdondogThe authors consider the numerical range and operator norm of \(V^2\), the square of the Voltera operator. Also, they get the numerical range, numerical radius and norm of the real (resp., imaginary) part of \(V^2\).
Reviewer: Mohammed El Aïdi (Bogotá)Some refinements of numerical radius inequalities for Hilbert space operatorshttps://zbmath.org/1496.470142022-11-17T18:59:28.764376Z"Alizadeh, Ebrahim"https://zbmath.org/authors/?q=ai:alizadeh.ebrahim"Farokhinia, Ali"https://zbmath.org/authors/?q=ai:farokhinia.aliSummary: The main goal of this paper is to obtain some refinements of numerical radius inequalities for Hilbert space operators.On the numerical range of some weighted shift operatorshttps://zbmath.org/1496.470152022-11-17T18:59:28.764376Z"Chakraborty, Bikshan"https://zbmath.org/authors/?q=ai:chakraborty.bikshan"Ojha, Sarita"https://zbmath.org/authors/?q=ai:ojha.sarita"Birbonshi, Riddhick"https://zbmath.org/authors/?q=ai:birbonshi.riddhickThe paper under review is devoted to computing the numerical radius of a weighted shift operator \(T\) on a complex Hilbert space with weights \((h,k,ab,ab,\ldots)\) where \(h,k,a,b\) are positive numbers and satisfy \(bh^2+(a+b)k^2>(a+b)^2b\). This is done viewing \(T\) as acting on a Hardy space \(H^2\) for appropriate weights related to \(T\), and studying the properties of an eigenfunction \(f\) of the operator \(\frac{T+T^*}{2}\) for an eigenvalue \(\alpha\) satisfying \(\alpha=\big\|\frac{T+T^*}{2}\big\|=w\big(\frac{T+T^*}{2}\big)=w(T)\). As an application, the authors obtain a lower bound of the numerical radius of some tridiagonal operators.
Reviewer: Javier Merí (Granada)Some new refinements of generalized numerical radius inequalities for Hilbert space operatorshttps://zbmath.org/1496.470162022-11-17T18:59:28.764376Z"Feki, Kais"https://zbmath.org/authors/?q=ai:feki.kais"Kittaneh, Fuad"https://zbmath.org/authors/?q=ai:kittaneh.fuadSummary: Let \(A\) be a positive (semi-definite) bounded linear operator on a complex Hilbert space \((\mathcal{H}, \langle \cdot, \cdot \rangle)\). Let \(\omega_A(T)\) and \(\Vert T\Vert_A\) denote the \(A\)-numerical radius and the \(A\)-operator seminorm of an operator \(T\) acting on the semi-Hilbert space \((\mathcal{H}, \langle \cdot, \cdot \rangle_A)\) respectively, where \(\langle x, y\rangle_A :=\langle Ax, y\rangle\) for all \(x, y\in\mathcal{H}\). It is well known that
\[
\frac{1}{4}\Vert T^{\sharp_A} T+TT^{\sharp_A}\Vert_A\le \omega_A^2(T) \le \frac{1}{2}\Vert T^{\sharp_A} T+TT^{\sharp_A}\Vert_A,
\]
where \(T^{\sharp_A}\) denotes a distinguished \(A\)-adjoint operator of \(T\). In this paper, we aim to give some new refinements of the above inequalities. Furthermore, we establish an \(\mathbb{A}\)-seminorm inequality involving \(2\times 2\) operator matrices, where \(\mathbb{A}=\mathrm{diag}(A, A)\). This generalizes a recent result of \textit{W. Bani-Domi} and \textit{F. Kittaneh} [Linear Multilinear Algebra 69, No. 5, 934--945 (2021; Zbl 07333203)]. As an application, a refinement of the triangle inequality related to \(\Vert \cdot \Vert_A\) is given.Some numerical radius inequalities for products of Hilbert space operatorshttps://zbmath.org/1496.470172022-11-17T18:59:28.764376Z"Hosseini, Mohsen Shah"https://zbmath.org/authors/?q=ai:hosseini.mohsen-shah"Moosavi, Baharak"https://zbmath.org/authors/?q=ai:moosavi.baharakSummary: We prove several numerical radius inequalities for products of two Hilbert space operators. Some of our inequalities improve well-known ones. More precisely, we prove that, if \(A,B \in \mathbb{B}(\mathscr{H})\) such that \(A\) is self-adjoint with \(\lambda_1 = \min \lambda_i \in \sigma (A)\) (the spectrum of \(A\)) and \(\lambda_2 = \max \lambda_i \in \sigma (A)\). Then
\[\omega(AB) \leq \|A\| \omega(B) + \left( \|A\| - \frac{| \lambda_1 + \lambda_2|}{2} \right) D_B\]
where \(D_B = \inf\limits_{\lambda \in \mathbb{C}} \|B - \lambda I \|\). In particular, if \( A>0\) and \( \sigma \subseteq [k\|A\|,\|A\|]\), then
\[\omega(AB) \leq (2-k) \|A\| \omega(B).\]Inverse continuity of the numerical range map for Hilbert space operatorshttps://zbmath.org/1496.470182022-11-17T18:59:28.764376Z"Lins, Brian"https://zbmath.org/authors/?q=ai:lins.brian"Spitkovsky, Ilya M."https://zbmath.org/authors/?q=ai:spitkovsky.ilya-matveyGiven a Hilbert space \(\mathcal{H}\) with inner product \(\langle\cdot,\cdot\rangle\), the numerical range \(W(A)\) of a bounded linear operator \(A\) on \(\mathcal{H}\) is the set
\[
W(A)=\{\langle Ax,x\rangle: x\in\mathcal{H}, \langle x,x\rangle=1\},
\]
while the numerical range map \(f_A\) associated with \(A\) is
\[
f_A: \{x\in\mathcal{H}: \langle x,x\rangle=1\}\rightarrow \mathbb{C},\quad x\mapsto \langle Ax,x\rangle.
\]
To describe the contents of the article, we quote its abstract:
``We describe continuity properties of the multivalued inverse of the numerical range map \(f_A: x\mapsto \langle Ax,x\rangle\) associated with a linear operator \(A\) defined on a complex Hilbert space \(\mathcal{H}\). We prove in particular that \(f_A^{-1}\) is strongly continuous at all points of the interior of the numerical range \(W(A)\). We give examples where strong and weak continuity fail on the boundary and address special cases such as normal and compact operators.''
Reviewer: Agnes Radl (Fulda)Almost invariant subspaces of the shift operator on vector-valued Hardy spaceshttps://zbmath.org/1496.470192022-11-17T18:59:28.764376Z"Chattopadhyay, Arup"https://zbmath.org/authors/?q=ai:chattopadhyay.arup"Das, Soma"https://zbmath.org/authors/?q=ai:das.soma"Pradhan, Chandan"https://zbmath.org/authors/?q=ai:pradhan.chandanThe authors characterize nearly invariant subspaces of finite defect for the backward shift operator acting on vector-valued Hardy spaces \(H^2_{\mathbb C^m}(\mathbb D)\), generalizing the scalar-valued result by \textit{I. Chalendar} et al. [J. Oper. Theory 83, No. 2, 321--331 (2020; Zbl 1463.47096)]. Given a bounded analytic function \(\Theta\) with values in the space of linear operators \(\mathcal L(\mathbb C^r,\mathbb C^m)\), we can induce the multiplier \(T_\theta F(z)=\Theta(z)F(z)\) from \(H^2_{\mathbb C^r}(\mathbb D)\) into \(H^2_{\mathbb C^m}(\mathbb D)\). They are determined by the condition \(ST_\Theta=T_\Theta S\) where \(S\) denotes the forward shift operator \(SF(z)=zF(z)\) which acts on the corresponding space in each case. We write, as usual, the backward shift \(S^* F(z)=\frac{F(z)-F(0)}{z}\). A closed subspace \(\mathcal M\subset H^2_{\mathbb C^m}(\mathbb D)\) is called almost-invariant for \(S\) if there exists a finite-dimensional subspace \(\mathcal F\subset H^2_{\mathbb C^m}(\mathbb D)\) such that \(S(\mathcal M)\subset \mathcal M\oplus\mathcal F\). Similarly, a closed subspace \(\mathcal M\subset H^2_{\mathbb C^m}(\mathbb D)\) is called nearly invariant for \(S^*\) if any \(F\in \mathcal M\) with \(F(0)=0\) satisfies that \(S^*F\in \mathcal M\) and it is called nearly \(S^*\)-invariant with defect \(p\) if there exists a \(p\)-dimensional subspace \(\mathcal F\subset H^2_{\mathbb C^m}(\mathbb D)\) such that, if \(F\in \mathcal M\) with \(F(0)=0\), then \(S^*F\in \mathcal M \oplus \mathcal F\). Also, \(\mathcal M\) is called \(S^*\)-almost invariant with defect \(p\) if \(S^*\mathcal M\subset\mathcal M \oplus \mathcal F\) and \(\dim \mathcal F=p\).
In the paper under review, the authors present a characterization of nearly invariant subspaces for \(S^*\) with finite defect in the vector-valued Hardy spaces. Using such a result, they also manage to obtain the description of almost invariant subspaces for the shift and its adjoint acting on vector-valued Hardy spaces.
Reviewer: Oscar Blasco (València)Diskcyclic \(C_0\)-semigroups and diskcyclicity criteriahttps://zbmath.org/1496.470202022-11-17T18:59:28.764376Z"Moosapoor, Mansooreh"https://zbmath.org/authors/?q=ai:moosapoor.mansoorehSummary: In this article, we prove that diskcyclic \(C_0\)-semigroups exist on any infinite-dimensional Banach space. We show that a \(C_0\)-semigroup \((T_t)_{t \geq 0}\) satisfies the diskcyclicity criterion if and only if any of \(T_t\)'s satisfies the diskcyclicity criterion for operators. Moreover, we show that there are diskcyclic \(C_0\)-semigroups that do not satisfy the diskcyclicity criterion. Also, we state various criteria for diskcyclicity of \(C_0\)-semigrous based on dense sets and \(d\)-dense orbits.Equivalence of semi-norms related to super weakly compact operatorshttps://zbmath.org/1496.470212022-11-17T18:59:28.764376Z"Tu, Kun"https://zbmath.org/authors/?q=ai:tu.kunLet $X$ and $Y$ be real infinite-dimensional Banach spaces. A subset $A$ of $X$ is said to be relatively super weakly compact if $A_{\mathcal{U}}$ is relatively weakly compact in $X_{\mathcal{U}}$ for any free ultrafilter $\mathcal{U}$. $A$ is said to be super weakly compact if it is weakly closed and relatively super weakly compact. The measure of super weak noncompactness of a bounded subset $A$ of $X$, $\sigma(A)$ is defined as
\[
\sigma(A)=\inf\{t > 0 : A\subset S + tB_X,\ S\text{ is relatively super weakly compact}\}.
\]
A bounded linear operator $T:X\to Y$ is called super weakly compact if $T(B_X)$ is relatively super weakly compact. Equivalently, $T_{\mathcal{U}}$ is weakly compact for any free ultrafilter $\mathcal{U}$. $A$ is called weakly compact if $T(B_X)$ is relatively weakly compact where $B_X$ is the closed unit ball in $X$.
Let $L(X,Y)$ denote the collection of all bounded linear operators mapping $X$ to $Y$ and $S(X,Y)$ represent the collection of all super weakly compact operators. The super weak essential norm $\|\cdot\|_s$ of $T\in L(X,Y)$ is the semi-norm induced from the quotient space $L(X,Y)/S(X,Y)$, that is,
\[
\|T\|_s=\inf\{\|T -S\|:S\in S(X,Y)\}.
\]
The space $X$ is said to have the super weakly compact approximation property (SWAP) if there is a real number $\lambda>0$ such that for any super weakly compact set $A\subset X$ and any $\varepsilon>0$, there is a super weakly compact operator $R:X\to X$ with $\sup_{x\in A} \|x- Rx\|\leq\varepsilon$ and $\|R\|\leq\lambda$.
In this paper, super weakly compact operators are discussed through a quantative method. By introducing the semi-norm $\sigma(T)$ of the operator $T:X\to Y$, which measures how far $T$ is from the family of super weakly compact operators, the following equivalence of the measure $\sigma(T)$ and the super weak essential norm $\|T\|_s$ of $T$ is proved:
Theorem. A Banach space $Y$ has the (SWAP) if only if the semi-norms $\sigma$ and ${\| \cdot\|}_s$ are equivalent in $L(X,Y)$ for any Banach space $X$.
In order to give an application of this theorem, some basic properties of Banach spaces having the SWAP are studied in Section~4 of the paper and then an example is constructed to show that the measures of $T$ and its dual $T^*$ are not always equivalent. Moreover, some examples of Banach spaces which have and which do not have the SWAP are given in this paper.
Reviewer: T. D. Narang (Amritsar)On two-dimensional model representations of one class of commuting operatorshttps://zbmath.org/1496.470222022-11-17T18:59:28.764376Z"Hatamleh, R."https://zbmath.org/authors/?q=ai:hatamleh.raed-m"Zolotarev, V. A."https://zbmath.org/authors/?q=ai:zolotarev.vladimir-alekseevichSummary: In the work by \textit{V. A. Zolotarev} [Dokl. Akad. Nauk Arm. SSR 63, No. 3, 136--140 (1976; Zbl 0351.47010)], a triangular model is constructed for a system of twice-commuting linear bounded completely nonself-adjoint operators \(\{A_1,A_2\}\) (\([A_1,A_2]=0\), \([A_1^\ast,A_2]=0\)) such that rank (\(A_1)_I(A_2)_I=1\) (\(2i(A_k)_I=A_k-A_k^\ast\), \(k=1,2\)) and the spectrum of each operator \(A_k\), \(k=1,2\), is concentrated at zero. The indicated triangular model has the form of a system of operators of integration over the independent variable in \(L_\Omega^2\) where the domain \(\Omega=[0,a]\times [0,b]\) is a compact set in \(\mathbb R^2\) bounded by the lines \(x=a\) and \(y=b\) and a decreasing smooth curve \(L\) connecting the points \((0,b)\) and \((a,0)\).Spectral theory for polynomially demicompact operatorshttps://zbmath.org/1496.470232022-11-17T18:59:28.764376Z"Brahim, Fatma Ben"https://zbmath.org/authors/?q=ai:brahim.fatma-ben"Jeribi, Aref"https://zbmath.org/authors/?q=ai:jeribi.aref"Krichen, Bilel"https://zbmath.org/authors/?q=ai:krichen.bilelSummary: In this article, we introduce the notion of polynomial demicompactness and we use it to give some results on Fredholm operators and to establish a fine description of some essential spectra of a closed densely defined linear operator. Our work is a generalization of many known ones in the literature.Essential pseudospectra involving demicompact and pseudodemicompact operators and some perturbation resultshttps://zbmath.org/1496.470242022-11-17T18:59:28.764376Z"Brahim, Fatma Ben"https://zbmath.org/authors/?q=ai:brahim.fatma-ben"Jeribi, Aref"https://zbmath.org/authors/?q=ai:jeribi.aref"Krichen, Bilel"https://zbmath.org/authors/?q=ai:krichen.bilelSummary: In this paper, we study the essential and the structured essential pseudospectra of closed densely defined linear operators acting on a Banach space \(X\). We start by giving a refinement and investigating the stability of these essential pseudospectra by means of the class of demicompact linear operators. Moreover, we introduce the notion of pseudo demicompactness and we study its relationship with pseudo upper semi-Fredholm operators. Some stability results for the Gustafson essential pseudospectrum involving pseudo demicompact operators is given.Property \((R)\) for functions of operators and its perturbationshttps://zbmath.org/1496.470252022-11-17T18:59:28.764376Z"Yang, Lili"https://zbmath.org/authors/?q=ai:yang.lili"Cao, Xiaohong"https://zbmath.org/authors/?q=ai:cao.xiaohongSummary: Let \(\mathcal{H}\) be a complex separable infinite dimensional Hilbert space and \(\mathcal{B(H)}\) be the algebra of all bounded linear operators on \(\mathcal{H}\). \(T\in \mathcal{B(H)}\) is said to satisfy property \((R)\) if \(\sigma_a(T)\backslash\sigma_{ab}(T)=\pi_{00}(T)\), where \(\sigma_a(T)\) and \(\sigma_{ab}(T)\) denote the approximate point spectrum and the Browder essential approximate point spectrum of \(T\), respectively, and \(\pi_{00}(T)=\{\lambda \in \operatorname{iso}\sigma (T): 0 < \dim N(T-\lambda I)<\infty\}\). In this paper, using a new spectrum, we talk about the property \((R)\) for functions of operators as well as its stability.Interpolating matriceshttps://zbmath.org/1496.470262022-11-17T18:59:28.764376Z"Dayan, Alberto"https://zbmath.org/authors/?q=ai:dayan.albertoThe main result is a matrix version of the Carleson theorem [\textit{L. Carleson}, Am. J. Math. 80, 921--930 (1958; Zbl 0085.06504)] which says that a sequence of points \(\Lambda=(\lambda_k)_{n\in\mathbb{N}}\) in the open unit disk \(\mathbb{D}\) is an interpolation sequence in \(H^\infty\) if and only if it is strongly separated. Here, \(\Lambda\) is replaced by a matrix sequence \(A=(A_n)_{n\in\mathbb{N}}\) (possibly with varying dimensions) with spectra in \(\mathbb{D}\) and interpolating means that for any bounded sequence \((\phi_n)_{n\in\mathbb{N}}\) in \(H^\infty\) there is a \(\varphi\in H^\infty\) such that \(\varphi(A_n)=\phi_n(A_n)\), \(n\in\mathbb{N}\). Strong separated means \(\inf_{z\in\mathbb{D}}\sup_{n\in\mathbb{N}}\prod_{k\ne n}|B_{A_k}(z)|>0\) in \(\mathbb{D}\) where \(B_M(z)\) is a Blaschke product whose zeros are eigenvalues of \(M\) (including multiplicity). An equivalent formulation is that the model spaces \(H_n=H^2\ominus B_{A_n}H^2\) are strongly separated. If the dimensions of the matrices \(A_n\) are uniformly bounded, then \((H_n)_{n\in\mathbb{N}}\) being a weakly separated Bessel system is an alternative equivalent condition for \(A\) to be interpolating.
Reviewer: Adhemar Bultheel (Leuven)New properties of the multivariable \(H^\infty\) functional calculus of sectorial operatorshttps://zbmath.org/1496.470272022-11-17T18:59:28.764376Z"Arrigoni, Olivier"https://zbmath.org/authors/?q=ai:arrigoni.olivier"Le Merdy, Christian"https://zbmath.org/authors/?q=ai:le-merdy.christianLet \(A : \operatorname{dom}A \rightarrow X\) be a closed and densely defined operator on a Banach space \(X\). We say that \(A\) is sectorial of type \(\omega \in(0, \pi)\) if its spectrum \(\sigma(A)\) satisfies \(\sigma(A) \subseteq \overline{\Sigma_\omega}\) and for any \(\theta \in (\omega, \pi)\), there exists a constant \(C_\theta \geq 0\) such that
\[
\|z R(z,A)\| \leq C_\theta \qquad (z \in \mathbb{C} \setminus \overline{\Sigma_\theta}),
\]
where \(R(z,A) = (z I_X - A)^{-1}\), and
\[
\Sigma_\theta = \{z \in \mathbb{C}^*: |\!\operatorname{Arg} z| < \theta\}.
\]
The paper under review concerns a multivariable \(H^\infty\)-functional calculus associated with commuting families of sectorial operators on Banach spaces.
One of the main results of this paper extends a result of \textit{N. J. Kalton} and \textit{L. Weis} [Math. Ann. 321, No. 2, 319--345 (2001; Zbl 0992.47005)] on the optimal angle of bounded \(H^\infty\)-functional calculus into the setting of tuples of operators. Another important result states: If \(X\) is reflexive and \(K\)-convex, then a tuple \((A_1, \dots, A_d)\) admitting a bounded \(H^\infty(\Sigma_{\theta_1} \times \dots \times \Sigma_{\theta_d})\) joint functional calculus for some \(\theta_k < \frac{\pi}{2}\), \(k=1, \ldots, d\), dilates into a commuting tuple of sectorial operators \((B_1, \dots, B_d)\) on a Bochner space admitting bounded \(H^\infty(\Sigma_{\frac{\pi}{2}} \times \dots \times \Sigma_{\frac{\pi}{2}})\) joint functional calculus, where each \(B_k\) generates a bounded \(C_0\)-group.
Reviewer: Jaydeb Sarkar (Bangalore)On a class of operator equations in locally convex spaceshttps://zbmath.org/1496.470282022-11-17T18:59:28.764376Z"Mishin, Sergeĭ N."https://zbmath.org/authors/?q=ai:mishin.sergey-nSummary: We consider a general method of solving equations whose left-hand side is a series by powers of a linear continuous operator acting in a locally convex space. Obtained solutions are given by operator series by powers of the same operator as the left-hand side of the equation. The research is realized by means of characteristics (of order and type) of operator as well as operator characteristics (of operator order and operator type) of vector relatively of an operator. In research we also use a convergence of operator series on equicontinuous bornology.Numerical interpretation of the Gurov-Reshetnyak inequality on the real axishttps://zbmath.org/1496.470292022-11-17T18:59:28.764376Z"Didenko, V. D."https://zbmath.org/authors/?q=ai:didenko.victor-d"Korenovskiĭ, A. A."https://zbmath.org/authors/?q=ai:korenovskii.anatolii-a"Tuah, N. J."https://zbmath.org/authors/?q=ai:tuah.nor-jaidiSummary: We find the ``norm'' of a power function in the Gurov-Reshetnyak class on the real line. Moreover, as a result of numerical experiments, we establish a lower bound for the norm of the operator of even extension of a function from the Gurov-Reshetnyak class from the semiaxis onto the entire real line.Applications of Kato's inequality for \(n\)-tuples of operators in Hilbert spaces. IIhttps://zbmath.org/1496.470302022-11-17T18:59:28.764376Z"Dragomir, Sever S."https://zbmath.org/authors/?q=ai:dragomir.sever-silvestru"Cho, Yeol Je"https://zbmath.org/authors/?q=ai:cho.yeol-je"Kim, Young-Ho"https://zbmath.org/authors/?q=ai:kim.youngho.1Summary: In this paper, by the use of famous Kato's inequality for bounded linear operators, we establish some new inequalities for \(n\)-tuples of operators and apply them to functions of normal operators defined by power series as well as to some norms and numerical radii that arise in multivariate operator theory. They provide a natural continuation of the results in Part I [\textit{S. S. Dragomir} et al., J. Inequal. Appl. 2013, Paper No. 21, 16 p. (2013; Zbl 1294.47028)].Čebyšev's type inequalities and power inequalities for the Berezin number of operatorshttps://zbmath.org/1496.470312022-11-17T18:59:28.764376Z"Garayev, Mubariz T."https://zbmath.org/authors/?q=ai:garayev.mubariz-tapdigoglu"Yamancı, Ulaş"https://zbmath.org/authors/?q=ai:yamanci.ulasSummary: We give operator analogues of some classical inequalities, including Čebyšev type inequality for synchronous and convex functions of selfadjoint operators in Reproducing Kernel Hilbert Spaces (RKHSs). We obtain some Berezin number inequalities for the product of operators. Also, we prove the Berezin number inequality for the commutator of two operators.Stability of versions of the Weyl-type theorems under the tensor producthttps://zbmath.org/1496.470322022-11-17T18:59:28.764376Z"Rashid, M. H. M."https://zbmath.org/authors/?q=ai:rashid.mohammad-hussein-mohammad|rashid.malik-h-m"Prasad, T."https://zbmath.org/authors/?q=ai:prasad.t-jayachandra|prasad.t-v-s-r-k|prasad.t-ram|prasad.tribhuan|prasad.t-b-arunaSummary: We study the transformation versions of the Weyl-type theorems for operators \(T\) and \(S\) and their tensor product \(T \otimes S\) in the infinite-dimensional space setting.Example of a quasianalytic contraction whose spectrum is a proper subarc of the unit circlehttps://zbmath.org/1496.470332022-11-17T18:59:28.764376Z"Gamal', Maria F."https://zbmath.org/authors/?q=ai:gamal.maria-fSummary: A partial answer to a question of \textit{L. Kérchy} and \textit{A. Szalai} [Proc. Am. Math. Soc. 143, No. 6, 2579--2584 (2015; Zbl 1321.47012)] is given. Namely, it is proved that there exists a quasianalytic contraction whose quasianalytic spectral set is equal to its spectrum and is a proper subarc of the unit circle, but no estimates of the norm of its inverse are given.Some properties of \((m, C)\)-isometric operatorshttps://zbmath.org/1496.470372022-11-17T18:59:28.764376Z"Li, Haiying"https://zbmath.org/authors/?q=ai:li.haiying"Wang, Yaru"https://zbmath.org/authors/?q=ai:wang.yaruSummary: In this paper, we study if \(T\) is an \((m,C)\)-isometric operator and \(CT^\ast C\) commutes with \(T\), then \(T^\ast\) is an \((m,C)\)-isometric operator. We also give local spectral properties and spectral relations of \((m,C)\)-isometric operators, such as property (\( \beta \)), decomposability, the single-valued extension property and Dunford's boundedness. We also investigate perturbation of \((m,C)\)-isometric operators by nilpotent operators and by algebraic operators and give some properties.Joint \(m\)-quasihyponormal operators on a Hilbert spacehttps://zbmath.org/1496.470382022-11-17T18:59:28.764376Z"Mahmoud, Sid Ahmed Ould Ahmed"https://zbmath.org/authors/?q=ai:sid-ahmed.ould-ahmed-mahmoud"Alshammari, Hadi Obaid"https://zbmath.org/authors/?q=ai:alshammari.hadi-obaidSummary: In this paper, We introduce a new class of multivariable operators known as joint \(m\)-quasihyponormal tuple of operators. It is a natural extension of joint normal and joint hyponormal tuples of operators. An \(m\)-tuple of operators \(\mathbf{S}=(S_1,\dots,S_m)\in\mathcal{B}(\mathcal{H})^m\) is said to be joint \(m\)-quasihyponormal tuple if \(\mathbf{S}\) satisfying
\[
\sum\limits_{1\le l,k\le m}\big \langle S_k^*\big [S_k^*,\;\; S_l\big ]S_lu_k\;|\;u_l\big \rangle \ge 0,
\]
for each finite collections \((u_l)_{1\le l\le m}\in\mathcal{H}\). Some properties of this class of multivariable operators are studied.Weyl-type theorems and \(k\)-quasi-\(M\)-hyponormal operatorshttps://zbmath.org/1496.470392022-11-17T18:59:28.764376Z"Zuo, Fei"https://zbmath.org/authors/?q=ai:zuo.fei"Zuo, Hongliang"https://zbmath.org/authors/?q=ai:zuo.hongliangSummary: In this paper, we show that if \(E\) is the Riesz idempotent for a non-zero isolated point \(\lambda\) of the spectrum of a \(k\)-quasi-\(M\)-hyponormal operator \(T\), then \(E\) is self-adjoint, and \(R(E) = N(T - \lambda) = N(T- \lambda)^\ast\). Also, we obtain that Weyl-type theorems hold for algebraically \(k\)-quasi-\(M\)-hyponormal operators.Essential spectrum of a weighted geometric realizationhttps://zbmath.org/1496.470422022-11-17T18:59:28.764376Z"Hatim, Khalid"https://zbmath.org/authors/?q=ai:hatim.khalid"Baalal, Azeddine"https://zbmath.org/authors/?q=ai:baalal.azeddineSummary: In this present article, we construct a new framework that's we call the weighted geometric realization of 2 and 3-simplexes. On this new weighted framework, we construct a nonself-adjoint 2-simplex Laplacian \(L\) and a self-adjoint 2-simplex Laplacian \(N\). We propose general conditions to ensure sectoriality for our new nonself-adjoint 2-simplex Laplacian \(L\). We show the relation between the essential spectra of \(L\) and \(N\). Finally, we prove the absence of the essential spectrum for our 2-simplex Laplacians \(L\) and \(N\).On couplings of symmetric operators with possibly unequal and infinite deficiency indiceshttps://zbmath.org/1496.470432022-11-17T18:59:28.764376Z"Mogilevskii, V. I."https://zbmath.org/authors/?q=ai:mogilevskii.vadimThe known results on couplings of symmetric operators \(A_j\), \(j\in\{1,2\}\), defined on the orthogonal sum of the Hilbert spaces, introduced by \textit{A. V. Shtraus} [Sov. Math., Dokl. 3, 779--782 (1962; Zbl 0151.19501); translation from Dokl. Akad. Nauk SSSR 144, 512--515 (1962)] are extended to the case of operators \(A_j\) with arbitrary (possibly unequal and infinite) deficiency indices. In particular, the coupling method based on the theory of boundary triplets is generalized. This makes it possible to obtain the abstract Titchmarsh formula, which gives the representation of the Weyl function of the coupling in terms of Weyl functions of boundary triplets for \(A_1\) and \(A_2\). \par Applications to ordinary differential operators are given.
Reviewer: Anatoly N. Kochubei (Kyïv)On double difference of composition operators from a space generated by the Cauchy kernel and a special measurehttps://zbmath.org/1496.470482022-11-17T18:59:28.764376Z"Sharma, Mehak"https://zbmath.org/authors/?q=ai:sharma.mehak"Sharma, Ajay K."https://zbmath.org/authors/?q=ai:sharma.ajay-kumar|sharma.ayay-k"Mursaleen, M."https://zbmath.org/authors/?q=ai:mursaleen.mohammad-ayman|mursaleen.mohammadSummary: In this paper, compact double difference of composition operators acting from a space generated by the Cauchy kernel and a special measure to analytic Besov spaces is characterized. Moreover, operator norm of these operators acting from Cauchy transforms to analytic Besov spaces is obtained explicitly.Reducing subspaces for a class of nonanalytic Toeplitz operatorshttps://zbmath.org/1496.470522022-11-17T18:59:28.764376Z"Deng, Jia"https://zbmath.org/authors/?q=ai:deng.jia"Lu, Yufeng"https://zbmath.org/authors/?q=ai:lu.yufeng"Shi, Yanyue"https://zbmath.org/authors/?q=ai:shi.yanyue"Hu, Yinyin"https://zbmath.org/authors/?q=ai:hu.yinyinSummary: In this paper, we give a uniform characterization for the reducing subspaces for \(T_{\varphi}\) with the symbol \(\varphi(z)=z^{k}+\bar{z}^{l}\) (\(k,l\in\mathbb{Z}_{+}^{2}\)) on the Bergman spaces over the bidisk, including the known cases that \(\varphi(z_{1},z_{2})=z_{1}^{N}z_{2}^{M}\) and \(\varphi(z_{1},z_{2})=z_{1}^{N}+\overline{z}_{2}^{M}\) with \(N,M\in\mathbb{Z}_{+}\). Meanwhile, the reducing subspaces for \(T_{z^{N}+\overline{z}^{M}}\) (\(N,M\in \mathbb{Z}_{+}\)) on the Bergman space over the unit disk are also described. Finally, we state these results in terms of the commutant algebra \(\mathcal{V}^{*}(\varphi)\).On the equivalence of some perturbations of the operator of multiplication by independent variablehttps://zbmath.org/1496.470612022-11-17T18:59:28.764376Z"Linchuk, Yu. S."https://zbmath.org/authors/?q=ai:linchuk.yu-sSummary: We study the conditions of equivalence of two operators obtained as perturbations of the operator of multiplication by independent variable by certain Volterra operators in the space of functions analytic in an arbitrary domain of the complex plane starlike with respect to the origin.Grüss-Landau inequalities for elementary operators and inner product type transformers in \(\mathrm{Q}\) and \(\mathrm{Q}^*\) norm ideals of compact operatorshttps://zbmath.org/1496.470622022-11-17T18:59:28.764376Z"Lazarević, Milan"https://zbmath.org/authors/?q=ai:lazarevic.milanSummary: For a probability measure $\mu$ on $\Omega$ and square integrable (Hilbert space) operator valued functions $\{A^*_t\}_{t\in \Omega}$, $\{B_t\}_{t\in\Omega}$, we prove Grüss-Landau type operator inequality for inner product type transformers
$$
\begin{multlined}
\left| \int_\Omega A_t X B_t \,d\mu(t) - \int_\Omega A_t\,d\mu(t) X \int_\Omega B_t \,d\mu(t) \right|^{2\eta} \\
\leqslant
\left\Vert \int_\Omega A_t A^*_t\,d\mu(t) - \left| \int_\Omega A^*_t \,d\mu(t) \right|^2 \right\Vert^\eta \left( \int_\Omega B^*_t X^* X B_t \,d\mu(t) - \left| X \int_\Omega B_t \,d\mu(t)\right|^2 \right)^\eta,
\end{multlined}
$$
for all $X \in \mathcal{B(H)}$ and for all $\eta \in [0,1]$.
Let $p\geqslant2$, $\Phi$ to be a symmetrically norming (s.n.) function, $\Phi^{(p)}$ to be its $p$-modification, $\Phi^{(p)^*}$ is a s.n. function adjoint to $\Phi^{(p)}$ and $\Vert\cdot\Vert_{\Phi^{(p)^*}}$ to be a norm on its associated ideal $\mathcal{C}_{\Phi^{(p)^*}}(\mathcal{H})$ of compact operators. If $X\in \mathcal{C}_{\Phi^{(p)^*}}(\mathcal{H})$ and $\{\alpha_n\}^\infty_{n=1}$ is a sequence in $(0,1]$, such that $\sum^\infty_{n=1}\alpha_n=1$ and $\sum^\infty_{n=1}\Vert\alpha ^{-1/2}_n A_n f \Vert^2 + \Vert \alpha^{-1/2}_n B^*_n f \Vert^2<+\infty$ for some families $\{A_n\}^\infty_{n=1}$ and $\{B_n\}^\infty_{n=1}$ of bounded operators on Hilbert space $\mathcal{H}$ and for all $f\in \mathcal{H}$, then
$$
\begin{multlined}
\left\Vert\sum^\infty_{n=1} \alpha^{-1}_n A_n X B_n - \sum^\infty_{n=1} A_n X \sum^\infty_{n=1} B_n \right\Vert _{\Phi^{(p)^*}} \\
\leqslant
\left\Vert \sqrt{ \sum^\infty_{n=1} \alpha^{-1}_n |A_n|^2 - \left| \sum^\infty_{n=1} A_n \right|^2} X \sqrt{ \sum^\infty_{n=1} \alpha^{-1}_n |B^*_n|^2 - \left| \sum^\infty_{n=1} B^*_n\right|^2} \right\Vert_{\Phi^{(p)^*}},
\end{multlined}
$$
if at least one of those operator families consists of mutually commuting normal operators.
The related Grüss-Landau type $\Vert\cdot\Vert_{\Phi^{(p)}}$ norm inequalities for inner product type transformers are also provided.Conditions of invertibility for functional operators with shift in weighted Hölder spaceshttps://zbmath.org/1496.470662022-11-17T18:59:28.764376Z"Tarasenko, G."https://zbmath.org/authors/?q=ai:tarasenko.george-s"Karelin, O."https://zbmath.org/authors/?q=ai:karelin.oleksandrSummary: We consider functional operators with shift in weighted Hölder spaces. The main result of the work is the proof of the conditions of invertibility for these operators. We also indicate the forms of the inverse operators. As an application, we propose to use these results for the solution of equations with shift encountered in the study of cyclic models for natural systems with renewable resources.On the \(C^\ast\)-algebra generated by the Bergman operator, Carleman second-order shift, and piecewise continuous coefficientshttps://zbmath.org/1496.471302022-11-17T18:59:28.764376Z"Mozel', V. A."https://zbmath.org/authors/?q=ai:mozel.v-aSummary: We study the \(C^\ast\)-algebra generated by the Bergman operator with piecewise continuous coefficients in the Hilbert space \(L_2\) and extended by the Carleman rotation by an angle \(\pi\). As a result, we obtain an efficient criterion for the operators from the indicated \(C^\ast\)-algebra to be Fredholm operators.Asymptotic properties of an optimal principal eigenvalue with spherical weight and Dirichlet boundary conditionshttps://zbmath.org/1496.490132022-11-17T18:59:28.764376Z"Ferreri, Lorenzo"https://zbmath.org/authors/?q=ai:ferreri.lorenzo"Verzini, Gianmaria"https://zbmath.org/authors/?q=ai:verzini.gianmariaSummary: We consider a weighted eigenvalue problem for the Dirichlet laplacian in a smooth bounded domain \(\Omega \subset \mathbb{R}^N\), where the bang-bang weight equals a positive constant \(\overline{m}\) on a ball \(B \subset \Omega\) and a negative constant \(- \underline{m}\) on \(\Omega \backslash B\). The corresponding positive principal eigenvalue provides a threshold to detect persistence/extinction of a species whose evolution is described by the heterogeneous Fisher-KPP equation in population dynamics. In particular, we study the minimization of such eigenvalue with respect to the position of \(B\) in \(\Omega \). We provide sharp asymptotic expansions of the optimal eigenpair in the singularly perturbed regime in which the volume of \(B\) vanishes. We deduce that, up to subsequences, the optimal ball concentrates at a point maximizing the distance from \(\partial \Omega \).Green function for gradient perturbation of unimodal Lévy processes in the real linehttps://zbmath.org/1496.601032022-11-17T18:59:28.764376Z"Grzywny, T."https://zbmath.org/authors/?q=ai:grzywny.tomasz"Jakubowski, T."https://zbmath.org/authors/?q=ai:jakubowski.tomasz"Żurek, G."https://zbmath.org/authors/?q=ai:zurek.grzegorzSummary: We prove that the Green function of the generator of symmetric unimodal Lévy process with the weak lower scaling order bigger than one and the Green functions of its gradient perturbations are comparable for bounded \(C^{1,1}\) subsets of the real line if the drift function is from an appropriate Kato class.Data smoothing with applications to edge detectionhttps://zbmath.org/1496.650572022-11-17T18:59:28.764376Z"Al-Jamal, Mohammad F."https://zbmath.org/authors/?q=ai:al-jamal.mohammad-f"Baniabedalruhman, Ahmad"https://zbmath.org/authors/?q=ai:baniabedalruhman.ahmad"Alomari, Abedel-Karrem"https://zbmath.org/authors/?q=ai:alomari.abedel-karremSummary: The aim of this paper is to present a new stable method for smoothing and differentiating noisy data defined on a bounded domain \(\Omega \subset \mathbb{R}^N\) with \(N\geq 1\). The proposed method stems from the smoothing properties of the classical diffusion equation; the smoothed data are obtained by solving a diffusion equation with the noisy data imposed as the initial condition. We analyze the stability and convergence of the proposed method and we give optimal convergence rates. One of the main advantages of this method lies in multivariable problems, where some of the other approaches are not easily generalized. Moreover, this approach does not require strong smoothness assumptions on the underlying data, which makes it appealing for detecting data corners or edges. Numerical examples demonstrate the feasibility and robustness of the method even with the presence of a large amounts of noise.Bound state solutions and thermodynamic properties of modified exponential screened plus Yukawa potentialhttps://zbmath.org/1496.810482022-11-17T18:59:28.764376Z"Antia, Akaninyene D."https://zbmath.org/authors/?q=ai:antia.akaninyene-d"Okon, Ituen B."https://zbmath.org/authors/?q=ai:okon.ituen-b"Isonguyo, Cecilia N."https://zbmath.org/authors/?q=ai:isonguyo.cecilia-n"Akankpo, Akaninyene O."https://zbmath.org/authors/?q=ai:akankpo.akaninyene-o"Eyo, Nsemeke E."https://zbmath.org/authors/?q=ai:eyo.nsemeke-eSummary: In this research paper, the approximate bound state solutions and thermodynamic properties of Schrödinger equation with modified exponential screened plus Yukawa potential (MESPYP) were obtained with the help Greene-Aldrich approximation to evaluate the centrifugal term. The Nikiforov-Uvarov (NU) method was used to obtain the analytical solutions. The numerical bound state solutions of five selected diatomic molecules, namely mercury hydride (HgH), zinc hydride (ZnH), cadmium hydride (CdH), hydrogen bromide (HBr) and hydrogen fluoride (HF) molecules were also obtained. We obtained the energy eigenvalues for these molecules using the resulting energy eigenequation and total unnormalized wave function expressed in terms of associated Jacobi polynomial. The resulting energy eigenequation was presented in a closed form and applied to study partition function (Z) and other thermodynamic properties of the system such as vibrational mean energy (U), vibrational specific heat capacity (C), vibrational entropy (S) and vibrational free energy (F). The numerical bound state solutions were obtained from the resulting energy eigenequation for some orbital angular quantum number. The results obtained from the thermodynamic properties are in excellent agreement with the existing literature. All numerical computations were carried out using spectroscopic constants of the selected diatomic molecules with the help of MATLAB 10.0 version. The numerical bound state solutions obtained increases with an increase in quantum state.Evolution of energy and magnetic moment of a quantum charged particle in power-decaying magnetic fieldshttps://zbmath.org/1496.810522022-11-17T18:59:28.764376Z"Dodonov, V. V."https://zbmath.org/authors/?q=ai:dodonov.victor-v"Horovits, M. B."https://zbmath.org/authors/?q=ai:horovits.m-bSummary: We consider a quantum spinless nonrelativistic charged particle moving in the \(xy\) plane under the action of a homogeneous time-dependent magnetic field \(B(t) = B_0(1 + t/t_0)^{-1-g}\), directed along the \(z\)-axis and described by means of the vector potential \(\mathbf{A}(t) = B(t)[-y, x]/2\). Assuming that the particle was initially in the thermal equilibrium state with a negligible coupling to a reservoir, we obtain exact formulas for the time-dependent mean values of the energy and magnetic moment in terms of the Bessel functions. Different regimes of the evolution are discovered and illustrated in several figures. The energy goes asymptotically to a finite value if \(g > 0\) (``fast'' decay), while it goes asymptotically to zero if \(g \leq 0\) (``slow'' decay). The dependence on parameter \(t_0\) practically disappears when \(1 + g\) is close to zero value (``superslow'' decay). The mean magnetic moment goes to zero for \(g > 1\), while it grows unlimitedly if \(g < 1\).Analysis of solutions of time-dependent Schrödinger equation of a particle trapped in a spherical boxhttps://zbmath.org/1496.810532022-11-17T18:59:28.764376Z"Nath, Debraj"https://zbmath.org/authors/?q=ai:nath.debraj"Carbó-Dorca, Ramon"https://zbmath.org/authors/?q=ai:carbo-dorca.ramonSummary: Three sets of exact solutions of the time-dependent Schrödinger equation of a particle that is trapped in a spherical box with a moving boundary wall have been calculated analytically. For these solutions, some physical quantities such as time-dependent average energy, average force, disequilibrium, quantum similarity measures as well as quantum similarity index have been investigated. Moreover, these solutions are compared concerning these physical quantities. The time-correlation functions among these solutions are investigated.On the number of eigenvalues of the lattice model operator in one-dimensional casehttps://zbmath.org/1496.810562022-11-17T18:59:28.764376Z"Bozorov, I. N."https://zbmath.org/authors/?q=ai:bozorov.i-n"Khurramov, A. M."https://zbmath.org/authors/?q=ai:khurramov.abdumazhid-molikovichSummary: It is considered a model operator \(h_{\mu}(k),k\in\mathbb{T}\equiv(-\pi,\pi]\), corresponding to the Hamiltonian of systems of two arbitrary quantum particles on a one-dimensional lattice with a special dispersion function that describes the transfer of a particle from one site to another interacting by a some short-range attraction potential \(v_{\mu}, \mu=(\mu_0,\mu_1,\mu_2,\mu_3)\in\mathbb{R}^4_+ \). The number of eigenvalues of the operator \(h_{\mu}(k),k\in\mathbb{T}\) depending on the energy of the particle interaction vector \(\mu\in\mathbb{R}^4_+\) and the total quasi-momentum \(k\in\mathbb{T}\) is studied.On a generalized central limit theorem and large deviations for homogeneous open quantum walkshttps://zbmath.org/1496.810602022-11-17T18:59:28.764376Z"Carbone, Raffaella"https://zbmath.org/authors/?q=ai:carbone.raffaella"Girotti, Federico"https://zbmath.org/authors/?q=ai:girotti.federico"Hernandez, Anderson Melchor"https://zbmath.org/authors/?q=ai:hernandez.anderson-melchorSummary: We consider homogeneous open quantum walks on a lattice with finite dimensional local Hilbert space and we study in particular the position process of the quantum trajectories of the walk. We prove that the properly rescaled position process asymptotically approaches a mixture of Gaussian measures. We can generalize the existing central limit type results and give more explicit expressions for the involved asymptotic quantities, dropping any additional condition on the walk. We use deformation and spectral techniques, together with reducibility properties of the local channel associated with the open quantum walk. Further, we can provide a large deviation principle in the case of a fast recurrent local channel and at least lower and upper bounds in the general case.Energy shift of a uniformly moving two-level atom through a thermal reservoirhttps://zbmath.org/1496.810722022-11-17T18:59:28.764376Z"Cai, Huabing"https://zbmath.org/authors/?q=ai:cai.huabing"Wang, Li-Gang"https://zbmath.org/authors/?q=ai:wang.ligangSummary: We investigate the implications of an atomic constant velocity in the energy shift of a two-level atom inside the thermal bath of a quantum scalar field, which is described by the Bose-Einstein distribution. The use of DDC formalism shows that the contribution of thermal fluctuations on the atomic level shifts depends on the atomic velocity and the temperature of the heat reservoir but the contribution of radiation reaction is totally insusceptible. The resulting energy shifts are analyzed and examined in detail under different circumstances. The atomic uniform linear motion always broadens the atomic level spacing in the limit of low temperature but narrows down it in the limit of high temperature. Our work clearly indicates that the moving heat reservoir shifts the atomic levels in a way quite different from that of the static one.The factorization method for inverse scattering by a two-layered cavity with conductive boundary conditionhttps://zbmath.org/1496.810922022-11-17T18:59:28.764376Z"Ye, Jianguo"https://zbmath.org/authors/?q=ai:ye.jianguo"Yan, Guozheng"https://zbmath.org/authors/?q=ai:yan.guozhengSummary: In this paper we consider the inverse scattering problem of determining the shape of a two-layered cavity with conductive boundary condition from sources and measurements placed on a curve inside the cavity. First, we show the well-posedness of the direct scattering problem by using the boundary integral equation method. Then, we prove that the factorization method can be applied to reconstruct the interface of the two-layered cavity from near-field data. Some numerical experiments are also presented to demonstrate the feasibility and effectiveness of the factorization method.Four-dimensional factorization of the fermion determinant in lattice QCDhttps://zbmath.org/1496.810942022-11-17T18:59:28.764376Z"Giusti, Leonardo"https://zbmath.org/authors/?q=ai:giusti.leonardo"Saccardi, Matteo"https://zbmath.org/authors/?q=ai:saccardi.matteoSummary: In the last few years it has been proposed a one-dimensional factorization of the fermion determinant in lattice QCD with Wilson-type fermions that leads to a block-local action of the auxiliary bosonic fields. Here we propose a four-dimensional generalization of this factorization. Possible applications are more efficient parallelizations of Monte Carlo algorithms and codes, master field simulations, and multi-level integration.Radiative (anti)neutrino energy spectra from muon, pion, and kaon decayshttps://zbmath.org/1496.811062022-11-17T18:59:28.764376Z"Tomalak, Oleksandr"https://zbmath.org/authors/?q=ai:tomalak.oleksandrSummary: To describe low-energy (anti)neutrino fluxes in modern coherent elastic neutrino-nucleus scattering experiments as well as high-energy fluxes in precision-frontier projects such as the Enhanced NeUtrino BEams from kaon Tagging (ENUBET) and the Neutrinos from STORed Muons (nuSTORM), we evaluate (anti)neutrino energy spectra from radiative muon (\(\mu^- \to e^- \bar{\nu}_e \nu_\mu(\gamma)\), \(\mu^+ \to e^+ \nu_e \bar{\nu}_\mu(\gamma)\)), pion \(\pi_{\ell2}\) (\(\pi^- \to \mu^- \bar{\nu}_\mu(\gamma)\), \(\pi^+ \to \mu^+ \nu_\mu(\gamma)\)), and kaon \(K_{\ell 2}\) (\(K^- \to \mu^- \bar{\nu}_\mu(\gamma)\), \(K^+ \to \mu^+ \nu_\mu(\gamma)\)) decays. We compare detailed \(\mathrm{O}(\alpha)\) distributions to the well-known tree-level results, investigate electron-mass corrections and provide energy spectra in analytical form. Radiative corrections introduce continuous and divergent spectral components near the endpoint, on top of the monochromatic tree-level meson-decay spectra, which can change the flux-averaged cross section at \(6 \times 10^{-5}\) level for the scattering on \(^{40}\mathrm{Ar}\) nucleus with (anti)neutrinos from the pion decay at rest. Radiative effects modify the expected (anti)neutrino fluxes from the muon decay around the peak region by 3--4 permille, which is a precision goal for next-generation artificial neutrino sources.Very special linear gravity: a gauge-invariant graviton masshttps://zbmath.org/1496.830012022-11-17T18:59:28.764376Z"Alfaro, Jorge"https://zbmath.org/authors/?q=ai:alfaro.jorge"Santoni, Alessandro"https://zbmath.org/authors/?q=ai:santoni.alessandroSummary: Linearized gravity in the Very Special Relativity (VSR) framework is considered. We prove that this theory allows for a non-zero graviton mass \(m_g\) without breaking gauge invariance nor modifying the relativistic dispersion relation. We find the analytic solution for the new equations of motion in our gauge choice, verifying as expected the existence of only two physical degrees of freedom. Finally, through the geodesic deviation equation, we confront some results for classic gravitational waves (GW) with the VSR ones: we see that the ratios between VSR effects and classical ones are proportional to \((m_g/E)^2\), \(E\) being the energy of a graviton in the GW. For GW detectable by the interferometers LIGO and VIRGO this ratio is at most \(10^{-20}\). However, for GW in the lower frequency range of future detectors, like LISA, the ratio increases significantly to \(10^{-10}\), that combined with the anisotropic nature of VSR phenomena may lead to observable effects.Möbius mirrorshttps://zbmath.org/1496.830062022-11-17T18:59:28.764376Z"Good, Michael R. R."https://zbmath.org/authors/?q=ai:good.michael-r-r"Linder, Eric V."https://zbmath.org/authors/?q=ai:linder.eric-vHow not to extract information from black holes: cosmic censorship as a guiding principlehttps://zbmath.org/1496.830272022-11-17T18:59:28.764376Z"Di Gennaro, Sofia"https://zbmath.org/authors/?q=ai:di-gennaro.sofia"Ong, Yen Chin"https://zbmath.org/authors/?q=ai:ong.yen-chinSummary: Black holes in general relativity are commonly believed to evolve towards a Schwarzschild state as they gradually lose angular momentum and electrical charge under Hawking evaporation. However, when Kim and Wen applied quantum information theory to Hawking evaporation and argued that Hawking particles with maximum mutual information could dominate the emission process, they found that charged black holes tend towards extremality. In view of some evidence pointing towards extremal black holes being effectively singular, this would violate the cosmic censorship conjecture. Nevertheless, since the Kim-Wen model is too simplistic (e.g. it assumes a continuous spectrum of particles with arbitrary charge-to-mass ratio), one might hope that a more realistic model could avoid this problem. In this work, we show that having only a finite species of charged particles would actually worsen the situation, with some end states becoming a naked singularity. With this model as an example, we emphasize the need to study whether charged black holes can violate cosmic censorship under a given model of Hawking evaporation.Laplacian on fuzzy de Sitter spacehttps://zbmath.org/1496.830332022-11-17T18:59:28.764376Z"Brkić, Bojana"https://zbmath.org/authors/?q=ai:brkic.bojana"Burić, Maja"https://zbmath.org/authors/?q=ai:buric.maja"Latas, Duško"https://zbmath.org/authors/?q=ai:latas.duskoUniformly accelerated Brownian oscillator in (2+1)D: temperature-dependent dissipation and frequency shifthttps://zbmath.org/1496.830352022-11-17T18:59:28.764376Z"Moustos, Dimitris"https://zbmath.org/authors/?q=ai:moustos.dimitrisSummary: We consider an Unruh-DeWitt detector modeled as a harmonic oscillator that is coupled to a massless quantum scalar field in the (2+1)-dimensional Minkowski spacetime. We treat the detector as an open quantum system and employ a quantum Langevin equation to describe its time evolution, with the field, which is characterized by a frequency-independent spectral density, acting as a stochastic force. We investigate a point-like detector moving with constant acceleration through the Minkowski vacuum and an inertial one immersed in a thermal reservoir at the Unruh temperature, exploring the implications of the well-known non-equivalence between the two cases on their dynamics. We find that both the accelerated detector's dissipation rate and the shift of its frequency caused by the coupling to the field bath depend on the acceleration temperature. Interestingly enough this is not only in contrast to the case of inertial motion in a heat bath but also to any analogous quantum Brownian motion model in open systems, where dissipation and frequency shifts are not known to exhibit temperature dependencies. Nonetheless, we show that the fluctuating-dissipation theorem still holds for the detector-field system and in the weak-coupling limit an accelerated detector is driven at late times to a thermal equilibrium state at the Unruh temperature.Stability analysis of anisotropic Bianchi type-I cosmological model in teleparallel gravityhttps://zbmath.org/1496.830362022-11-17T18:59:28.764376Z"Koussour, M."https://zbmath.org/authors/?q=ai:koussour.m"Bennai, M."https://zbmath.org/authors/?q=ai:bennai.mohamedElectromagnetic effects on dynamics of string fluid and information paradox in rainbow gravityhttps://zbmath.org/1496.830372022-11-17T18:59:28.764376Z"Sheikh, Umber"https://zbmath.org/authors/?q=ai:sheikh.umber"Arshad, Sana"https://zbmath.org/authors/?q=ai:arshad.sanaSummary: This work is devoted to studying the effects of electric field intensity on collapsing anisotropic string fluid in Rainbow gravity. The Einstein field equations are modified and solved for the spherical symmetric spacetime. The physical parameters of fluid including energy density, pressure and string tension are obtained. Moreover, the time and radius of formation of the apparent horizon are estimated. All these quantities depend on the fluid's electric intensity. The graphical analysis of the physical existence of dynamical quantities depending on the energy of the probing particle is presented. It is found that the presence of an electric field decreases the mass density and increases fluid's pressure. The electric field increases the time and radius of apparent horizon formation resulting in slowing down the collapsing process.Interaction of inhomogeneous axions with magnetic fields in the early universehttps://zbmath.org/1496.830382022-11-17T18:59:28.764376Z"Dvornikov, Maxim"https://zbmath.org/authors/?q=ai:dvornikov.maximSummary: We study the system of interacting axions and magnetic fields in the early universe after the quantum chromodynamics phase transition, when axions acquire masses. Both axions and magnetic fields are supposed to be spatially inhomogeneous. We derive the equations for the spatial spectra of these fields, which depend on conformal time. In case of the magnetic field, we deal with the spectra of the energy density and the magnetic helicity density. The evolution equations are obtained in the closed form within the mean field approximation. We choose the parameters of the system and the initial condition which correspond to realistic primordial magnetic fields and axions. The system of equations for the spectra is solved numerically. We compare the cases of inhomogeneous and homogeneous axions. The evolution of the magnetic field in these cases is different only within small time intervals. Generally, magnetic fields are driven mainly by the magnetic diffusion. We find that the magnetic field instability takes place for the amplified initial wavefunction of the homogeneous axion. This instability is suppressed if we account for the inhomogeneity of the axion.Emergence of space and expansion of universehttps://zbmath.org/1496.830402022-11-17T18:59:28.764376Z"V T, Hassan Basari"https://zbmath.org/authors/?q=ai:v-t.hassan-basari"Krishna, P. B."https://zbmath.org/authors/?q=ai:krishna.p-b"K. V, Priyesh"https://zbmath.org/authors/?q=ai:k-v.priyesh"Mathew, Titus K."https://zbmath.org/authors/?q=ai:mathew.titus-kQuark condensate and chiral symmetry restoration in neutron starshttps://zbmath.org/1496.850012022-11-17T18:59:28.764376Z"Jin, Hao-Miao"https://zbmath.org/authors/?q=ai:jin.hao-miao"Xia, Cheng-Jun"https://zbmath.org/authors/?q=ai:xia.cheng-jun"Sun, Ting-Ting"https://zbmath.org/authors/?q=ai:sun.tingting"Peng, Guang-Xiong"https://zbmath.org/authors/?q=ai:peng.guang-xiongSummary: Based on an equivparticle model, we investigate the in-medium quark condensate in neutron stars. Carrying out a Taylor expansion of the nuclear binding energy to the order of \(\rho^3\), we obtain a series of EOSs for neutron star matter, which are confronted with the latest nuclear and astrophysical constraints. The in-medium quark condensate is then extracted from the constrained properties of neutron star matter, which decreases non-linearly with density. However, the chiral symmetry is only partially restored with non-vanishing quark condensates, which may vanish at a density that is out of reach for neutron stars.