Recent zbMATH articles in MSC 47Ahttps://zbmath.org/atom/cc/47A2022-09-13T20:28:31.338867ZWerkzeugAlmost commuting matrices with respect to the rank metrichttps://zbmath.org/1491.150222022-09-13T20:28:31.338867Z"Elek, Gábor"https://zbmath.org/authors/?q=ai:elek.gabor"Grabowski, Łukasz"https://zbmath.org/authors/?q=ai:grabowski.lukaszThis article discusses a variant of the famous Halmos problem: to perturb two almost commuting matrices such that their approximate versions are commuting. In general, not all such matrix perturbations exist.
The answer to the Halmos problem is positive when the two given matrices are self-adjoint and with norm one. However, the answer is negative when the two matrices are unitary. Using the different distance metrics, such as the Hamming distance or the normalized Hilbert-Schmidt norm for defining the closeness of two matrices, the answer to the above problem is also positive, as proved by \textit{P. Rosenthal} [Amer. Math. Monthly 76, No. 8, 925--926 (1969)], \textit{G. Arzhantseva} and \textit{L. Păunescu} [J. Funct. Anal. 269, No. 3, 745--757 (2015; Zbl 1368.20025)].
Instead of using the usually norm-induced metric, the present authors propose a rank metric to measure the closeness of a matrix (unitary or self-adjoint) and its perturbed version.
Like the Hamming distance, the rank distances have been used widely in coding theory. This paper might find applications in error-correcting codes, e.g., rank codes, which attracted the attention partly thanks to the work by \textit{È. M. Gabidulin} [Probl. Inf. Transm. 21, 1--12 (1985; Zbl 0585.94013); translation from Probl. Peredachi Inf. 21, No. 1, 3--16 (1985)]. Also interesting is the conclusion that Abel's group is not stable with the rank metric. The result might serve as a reminder that more attention should be taken when constructing well-behaved rank codes.
Reviewer: Liang Liao (London)Norm estimations for the Moore-Penrose inverse of the weak perturbation of matriceshttps://zbmath.org/1491.150272022-09-13T20:28:31.338867Z"Fu, Chunhong"https://zbmath.org/authors/?q=ai:fu.chunhong"Song, Chuanning"https://zbmath.org/authors/?q=ai:song.chuanning"Wang, Guorong"https://zbmath.org/authors/?q=ai:wang.guorong"Xu, Qingxiang"https://zbmath.org/authors/?q=ai:xu.qingxiangSummary: A multiplicative perturbation \(M\) of a matrix \(T\) has the form \(M=ETF^\ast\), where \(E\) and \(F\) are square matrices. It is proved that every acute perturbation is essentially a strong perturbation, which is a type of multiplicative perturbation. It is also proved that for every multiplicative perturbation \(M\), \(M\) is a strong perturbation if and only if it is a weak perturbation and is rank-preserving. Some norm equations for the Moore-Penrose inverse are derived in the framework of the weak perturbation, through which some norm upper bounds for \(M^\dag - T^\dag\) are obtained. As an application, the perturbation estimation for the solution to the least squares problems is provided. The sharpness of the newly obtained upper bounds are illustrated by several numerical examples.Linear operators preserving combinatorial matrix setshttps://zbmath.org/1491.150312022-09-13T20:28:31.338867Z"Shteyner, P. M."https://zbmath.org/authors/?q=ai:shteyner.p-mSummary: The paper investigates linear functionals \(\varphi : \mathbb{R}^n \rightarrow \mathbb{R}\) preserving a set \(\mathcal{M} \subseteq \mathbb{R} \), i.e., \( \varphi : \mathbb{R}^n \rightarrow \mathbb{R}\) such that \(\varphi (v) \in \mathcal{M}\) for any vector \(u \in \mathbb{R}^n\) with components in \(\mathcal{M} \). For various types of subsets of real numbers, characterizations of the linear functionals that preserve them are obtained. In particular, the sets \(\mathbb{Z},\mathbb{Q}, \mathbb{Z}_+, \mathbb{Q}_+, \mathbb{R}_+\), several infinite sets of integers, bounded and unbounded intervals, and finite subsets of real numbers are considered. A characterization of linear functionals preserving a set \(\mathcal{M}\) also allows one to describe the linear operators preserving matrices with entries from this set, i.e., the operators \(\Phi : M_{ n,m } \rightarrow M_{ n,m }\) such that all entries of the matrix \(\Phi (A)\) belong to \(\mathcal{M}\) for any matrix \(A \in M_{n,m}\) with all entries in \(\mathcal{M} \). As an example, linear operators preserving \((0, 1)\)-, \(( \pm 1)\)-, and \(( \pm 1, 0)\)-matrices are characterized.Alpha Procrustes metrics between positive definite operators: a unifying formulation for the Bures-Wasserstein and Log-Euclidean/Log-Hilbert-Schmidt metricshttps://zbmath.org/1491.150372022-09-13T20:28:31.338867Z"Minh, Hà Quang"https://zbmath.org/authors/?q=ai:minh.ha-quangSummary: This work presents a parametrized family of distances, namely the Alpha Procrustes distances, on the set of symmetric, positive definite (SPD) matrices. The Alpha Procrustes distances provide a unified formulation encompassing both the Bures-Wasserstein and Log-Euclidean distances between SPD matrices. We show that the Alpha Procrustes distances are the Riemannian distances corresponding to a family of Riemannian metrics on the manifold of SPD matrices, which encompass both the Log-Euclidean and Wasserstein Riemannian metrics. This formulation is then generalized to the set of positive definite Hilbert-Schmidt operators on a Hilbert space, unifying the infinite-dimensional Bures-Wasserstein and Log-Hilbert-Schmidt distances. In the setting of reproducing kernel Hilbert spaces (RKHS) covariance operators, we obtain closed form formulas for all the distances via the corresponding kernel Gram matrices. From a statistical viewpoint, the Alpha Procrustes distances give rise to a parametrized family of distances between Gaussian measures on Euclidean space, in the finite-dimensional case, and separable Hilbert spaces, in the infinite-dimensional case, encompassing the 2-Wasserstein distance, with closed form formulas via Gram matrices in the RKHS setting. The presented formulations are new both in the finite and infinite-dimensional settings.Random perturbations of matrix polynomialshttps://zbmath.org/1491.150422022-09-13T20:28:31.338867Z"Pagacz, Patryk"https://zbmath.org/authors/?q=ai:pagacz.patryk"Wojtylak, Michał"https://zbmath.org/authors/?q=ai:wojtylak.michalAuthors' abstract: A sum of a large-dimensional random matrix polynomial and a fixed low-rank matrix polynomial is considered. The main assumption is that the resolvent of the random polynomial converges to some deterministic limit. A formula for the limit of the resolvent of the sum is derived, and the eigenvalues are localised. Four instances are considered: a low-rank matrix perturbed by the Wigner matrix, a product \(HX\) of a fixed diagonal matrix \(H\) and the Wigner matrix \(X\) and two special matrix polynomials of higher degree. The results are illustrated with various examples and numerical simulations.
Reviewer: Sen Zhu (Changchun)Fractal convolution on the rectanglehttps://zbmath.org/1491.280112022-09-13T20:28:31.338867Z"Pasupathi, R."https://zbmath.org/authors/?q=ai:pasupathi.r"Navascués, M. A."https://zbmath.org/authors/?q=ai:navascues.maria-antonia"Chand, A. K. B."https://zbmath.org/authors/?q=ai:chand.arya-kumar-bedabrataThe authors prime objective with this paper is the investigation of fractal bases and frames for the Lebesgue space \(L^2(I\times J)\) where \(I\) and \(J\) are compact intervals of \(\mathbb{R}\). In particular, the (left and right) fractal convolution operator is used to derive results such as the existence of Bessel sequences, Riesz bases and frames consisting of products of self-referential functions.
Reviewer: Peter Massopust (München)Embedded eigenvalues of the Neumann problem in a strip with a box-shaped perturbationhttps://zbmath.org/1491.351322022-09-13T20:28:31.338867Z"Cardone, Giuseppe"https://zbmath.org/authors/?q=ai:cardone.giuseppe"Durante, T."https://zbmath.org/authors/?q=ai:durante.tiziana"Nazarov, S. A."https://zbmath.org/authors/?q=ai:nazarov.sergei-aleksandrovichSummary: We consider the spectral Neumann problem for the Laplace operator in an acoustic waveguide \(\Pi_l^\varepsilon\) formed by the union of an infinite strip and a narrow box-shaped perturbation of size \(2l\times\varepsilon\), where \(\varepsilon>0\) is a small parameter. We prove the existence of the length parameter \(l_k^\varepsilon=\pi k+O(\varepsilon)\) with any \(k=1,2,3,\dots\) such that the waveguide \(\Pi_{l_k^\varepsilon}^\varepsilon\) supports a trapped mode with an eigenvalue \(\lambda_k^\varepsilon=\pi^2-4\pi^4 l^2\varepsilon^2+O(\varepsilon^3)\) embedded into the continuous spectrum. This eigenvalue is unique in the segment \([0,\pi^2]\), and it is absent in the case \(l\neq l_k^\varepsilon\). The detection of this embedded eigenvalue is based on a criterion for trapped modes involving an artificial object, the augmented scattering matrix. The main difficulty is caused by the rather specific shape of the perturbed wall \(\partial\Pi_l^\varepsilon\), namely a narrow rectangular bulge with corner points, and we discuss available generalizations for other piecewise smooth boundaries.Existence of solutions for a nonlocal reaction-diffusion equation in biomedical applicationshttps://zbmath.org/1491.351432022-09-13T20:28:31.338867Z"Leon, Cristina"https://zbmath.org/authors/?q=ai:leon.cristina"Kutsenko, Irina"https://zbmath.org/authors/?q=ai:kutsenko.irina"Volpert, Vitaly"https://zbmath.org/authors/?q=ai:volpert.vitaly-aSummary: The paper is devoted to a nonlocal semi-linear elliptic equation in \(\mathbb{R}^n\) arising in various biological and biomedical applications. The Fredholm property studied for the corresponding linear elliptic operators with discontinuous coefficients allows the application of the implicit function theorem to prove the persistence of solutions under a small perturbation of the problem. Furthermore, the existence of solutions is established by the Leray-Schauder method based on the topological degree for Fredholm and proper operators and on a priori estimates of solutions in some special weighted spaces.Fredholm property and essential spectrum of 3-D Dirac operators with regular and singular potentialshttps://zbmath.org/1491.351502022-09-13T20:28:31.338867Z"Rabinovich, Vladimir"https://zbmath.org/authors/?q=ai:rabinovich.vladimir-l|rabinovich.vladimir-sIn the paper, the 3D-Dirac operators with singular potentials supported on both bounded and unbounded surfaces in \(\mathbb{R}^3\) are considered. The approach to the self-adjointness of Dirac operators is based on the study of transmission problems with parameter associated with the Dirac operators. For their invertibility for large values of the parameter, and for the a priori estimates of solutions to associated transmission problems an analogue of Lopatinsky conditions is introduced. Finally, the Fredholm properties and the essential spectrum of transmission problems associated with the Dirac operators with singular potentials with supports on compact surfaces and non-compact surfaces with conical exits to infinity are investigated.
Reviewer: David Kapanadze (Tbilisi)On a boundary problem for a fourth-order elliptic equation on a planehttps://zbmath.org/1491.351682022-09-13T20:28:31.338867Z"Soldatov, A. P."https://zbmath.org/authors/?q=ai:soldatov.aleksandr-pavlovichSummary: The paper consideres a boundary value problem for a fourth-order elliptic equation with constant real coefficients in a multiply connected domain, in which the function and its normal third-order derivative on the boundary of this domain are specified. A convenient Fredholmity criterion is given, and a formula for the index of this problem is presented. Classes of equations for which the Fredholmity criterion is especially simple are indicated, and the exact values of the index are calculated.The massless Dirac equation in two dimensions: zero-energy obstructions and dispersive estimateshttps://zbmath.org/1491.353742022-09-13T20:28:31.338867Z"Erdoğan, M. Burak"https://zbmath.org/authors/?q=ai:erdogan.mehmet-burak"Goldberg, Michael"https://zbmath.org/authors/?q=ai:goldberg.michael-joseph"Green, William R."https://zbmath.org/authors/?q=ai:green.william-rSummary: We investigate \(L^1\to L^\infty\) dispersive estimates for the massless two dimensional Dirac equation with a potential. In particular, we show that the Dirac evolution satisfies the natural \(t^{-\frac{1}{2}}\) decay rate, which may be improved to \(t^{-\frac{1}{2} - \gamma}\) for any \(0\leq \gamma < \frac{3}{2}\) at the cost of spatial weights. We classify the structure of threshold obstructions as being composed of a two dimensional space of p-wave resonances and a finite dimensional space of eigenfunctions at zero energy. We show that, in the presence of a threshold resonance, the Dirac evolution satisfies the natural decay rate except for a finite-rank piece. While in the case of a threshold eigenvalue only, the natural decay rate is preserved. In both cases we show that the decay rate may be improved at the cost of spatial weights.Inverse scattering for three-dimensional quasi-linear biharmonic operatorhttps://zbmath.org/1491.354572022-09-13T20:28:31.338867Z"Harju, Markus"https://zbmath.org/authors/?q=ai:harju.markus"Kultima, Jaakko"https://zbmath.org/authors/?q=ai:kultima.jaakko"Serov, Valery"https://zbmath.org/authors/?q=ai:serov.valery-sSummary: We consider an inverse scattering problem of recovering the unknown coefficients of a quasi-linearly perturbed biharmonic operator in the three-dimensional case. These unknown complex-valued coefficients are assumed to satisfy some regularity conditions on their nonlinearity, but they can be discontinuous or singular in their space variable. We prove Saito's formula and uniqueness theorem of recovering some essential information about the unknown coefficients from the knowledge of the high frequency scattering amplitude.On certain operator familieshttps://zbmath.org/1491.354742022-09-13T20:28:31.338867Z"Vasilyev, V. B."https://zbmath.org/authors/?q=ai:vasilyev.vladimir-bSummary: In this paper, we propose an abstract scheme for the study of special operators and apply this scheme to examining elliptic pseudo-differential operators and related boundary-value problems on manifolds with nonsmooth boundaries. In particular, we consider cases where boundaries may contain conical points, edges of various dimensions, and even peak points. Using the constructions proposed, we present well-posed formulations of boundary-value problems for elliptic pseudo-differential equations on manifolds discussed in Sobolev-Slobodecky spaces.The ergodic decomposition defined by actions of amenable groups. II: More about the role played by the ergodic invariant probability measures in the decompositionhttps://zbmath.org/1491.370062022-09-13T20:28:31.338867Z"Zaharopol, Radu"https://zbmath.org/authors/?q=ai:zaharopol.raduSummary: Our purpose here is to continue the study of the ergodic decomposition for actions defined by amenable groups, started in [the author, Colloq. Math. 165, No. 2, 285--319 (2021; Zbl 1478.37006)]. We consider the set \(\varGamma^{(w)}_{\alpha \mathrm{cpie}}\) defined in the above-mentioned paper, and we prove that it is Borel measurable and of maximal probability.The essence of invertible frame multipliers in scalabilityhttps://zbmath.org/1491.420442022-09-13T20:28:31.338867Z"Javanshiri, Hossein"https://zbmath.org/authors/?q=ai:javanshiri.hossein"Abolghasemi, Mohammad"https://zbmath.org/authors/?q=ai:abolghasemi.mohammad"Arefijamaal, Ali Akbar"https://zbmath.org/authors/?q=ai:arefijamaal.ali-akbarA separable Hilbert space is denoted by \(\mathcal{H}.\) A Bessel operator is an operator of the form \[ M_{m, \phi, \psi} (f) = \sum_{i = 1}^{\infty} m_i \langle f, \psi_i \rangle \phi_i, \quad f \in \mathcal{H}, \] where \(\phi\) and \(\psi\) are Bessel sequences in \(\mathcal{H}\), and \(m\) is a bounded complex scalar sequence in \(\mathbb{C}\). The first result (Proposition 1) in this paper pertains to the invertibility of a Bessel operator associated with a given sequence of nonzero scalars \(m\) and a frame \(\Phi = \{\phi_n\}\) of \(\mathcal{H}\). Proposition 1 is then applied to obtain several results on scalability of frames. Scalability of frames addresses the question of when, for a given frame \(\Phi\), there exist scalars \(\{c_n\}\), \(c_n \geq 0\), such that \(\{c_n \phi_n \}\) is a tight frame. If \(c_n > 0\) for all \(n\), then \(\Phi\) is called positively scalable, and if \(\textrm{inf}_n c_n > 0\) then \(\Phi\) is called strictly scalable.
\par The authors have shown that positive and strict scalability coincide for all frames \(\{ \phi_n \}_n\) with \(\lim \textrm{inf}_n \| \phi_n \| > 0\) which, in particular, provides some equivalent conditions for positive scalability of certain frames. In the process, the authors were able to completely characterize the scalability of Riesz bases and Riesz frames. Further, in the context of scalability of frames, by using Feichtinger Conjecture, the authors give \(\alpha\) and \(\beta\) such that the elements of the scaling sequence \(\{c_n\}\) should be chosen from the interval \([\alpha, \beta]\) for all but finitely many \(n.\) In their result, the values of \(\alpha\) and \(\beta\) depend on the optimal frame bounds of the frame \(\Phi\). The last result in the paper is of independent interest and gives an explicit algorithm to construct desired invertible multipliers from the given one.
Reviewer: Somantika Datta (Moscow)Invertibility of generalized \(g\)-frame multipliers in Hilbert spaceshttps://zbmath.org/1491.420452022-09-13T20:28:31.338867Z"Moosavianfard, Z."https://zbmath.org/authors/?q=ai:moosavianfard.z"Abolghasemi, M."https://zbmath.org/authors/?q=ai:abolghasemi.mahdi|abolghasemi.mohammad"Tolooei, Y."https://zbmath.org/authors/?q=ai:tolooei.yaserIn the present article, the authors discussed and studied \(g\)-frames in Hilbert spaces. The authors characterized the sequences which construct invertible generalized multipliers with a given \(g\)-Bessel sequence with an operator, \(g\)-frames with a bounded operator and \(g\)-frames with a semi-normalized operator. Necessary and sufficient conditions for a \(g\)-frame to be a \(g\)-Riesz basis are given. Some results to produce unique dual \(g\)-frames of \(g\)-frames are provided. The authors also discussed and constructed invertible generalized multipliers and obtained few results.
Reviewer: Virender Dalal (Delhi)\(K\)-frames from finite extensionshttps://zbmath.org/1491.420462022-09-13T20:28:31.338867Z"Rahmani, Morteza"https://zbmath.org/authors/?q=ai:rahmani.morteza.1|rahmani.mortezaGiven a bounded linear operator \(K\) on a separable Hilbert space \(H\), a sequence \(\{f_n\}_{n\in J}\) is called a \(K\)-frame for \(H\) if there exist constants \(A, B >0\) such that \[A\|K^*f\|^2\le \sum_{n\in J} |\langle f,f_n\rangle|^2 \le B\|f\|^2 \] for any \(f\in H\). The sequence is called Bessel if the second of the two inequalities above holds.
The author provides several results on the extension of Bessel sequences to \(K\)-frames. For example, necessary and sufficient conditions are given for a Bessel sequence to be extendable to a \(K\)-frame by a finite number of elements of \(H\). The conditions are presented in terms of the operator \(K\) and the analysis operator of the Bessel sequence and use classical characterizations of Fredholm operators.
Reviewer: Ilya Krishtal (DeKalb)Tracial moment problems on hypercubeshttps://zbmath.org/1491.440062022-09-13T20:28:31.338867Z"Le, Cong Trinh"https://zbmath.org/authors/?q=ai:le-cong-trinh.Summary: In this paper we introduce the \textit{tracial \(K\)-moment problem} and the \textit{sequential matrix-valued \(K\)-moment problem} and show the equivalence of the solvability of these problems. Using a Haviland's theorem for matrix polynomials, we solve these \(K\)-moment problems for the case where \(K\) is the hypercube \([-1,1]^n\).The resolvent and the particular solution of one singular integral equationhttps://zbmath.org/1491.450022022-09-13T20:28:31.338867Z"Jenaliyev, Muvasharkhan"https://zbmath.org/authors/?q=ai:dzhenaliev.muvasharkhan-t"Iskakov, Sagyndyk"https://zbmath.org/authors/?q=ai:iskakov.sagyndyk"Ramazanov, Murat"https://zbmath.org/authors/?q=ai:ramazanov.murat-ibraevichSummary: In this paper, we study the solvability of a singular integral equation arising in the theory of boundary value problems for the heat equation in an infinite angular domain. With the help of thermal potentials, the boundary value problem of heat conduction is reduced to a singular integral Volterra equation of the second kind. The corresponding homogeneous integral equation was investigated by us earlier in the previous paper of [\textit{M. T. Jenaliyev} and \textit{M. I. Ramazanov}, Mat. Zh. 16, No. 4, 60--76 (2016; Zbl 1488.35268)], and it was shown that in some weight class of essentially bounded functions it has, along with a trivial solution, a family of nontrivial solutions up to a constant factor. In this paper, we study the nonhomogeneous integral equation, for which a representation of the general solution is found with the help of the resolvent constructed by us. Estimates of the resolvent are established.
For the entire collection see [Zbl 1436.46003].On the factorable spaces of absolutely \(p\)-summable, null, convergent, and bounded sequenceshttps://zbmath.org/1491.460072022-09-13T20:28:31.338867Z"Başar, Feyzi"https://zbmath.org/authors/?q=ai:basar.feyzi"Roopaei, Hadi"https://zbmath.org/authors/?q=ai:roopaei.hadiSummary: Let \(F\) denote the factorable matrix and \(X\in\{ \ell_p,c_0,c, \ell_\infty\}\). In this study, we introduce the domains \(X(F)\) of the factorable matrix in the spaces \(X\). Also, we give the bases and determine the alpha-, beta- and gamma-duals of the spaces \(X(F)\). We obtain the necessary and sufficient conditions on an infinite matrix belonging to the classes \((\ell_p(F),\ell_\infty)\), \((\ell_p(F),f)\) and \((X,Y(F))\) of matrix transformations, where \(Y\) denotes any given sequence space. Furthermore, we give the necessary and sufficient conditions for factorizing an operator based on the matrix \(F\) and derive two factorizations for the Cesàro and Hilbert matrices based on the Gamma matrix. Additionally, we investigate the norm of operators on the domain of the matrix \(F\). Finally, we find the norm of Hilbert operators on some sequence spaces and deal with the lower bound of operators on the domain of the factorable matrix.Central amalgamation of groups and the RFD propertyhttps://zbmath.org/1491.460522022-09-13T20:28:31.338867Z"Shulman, Tatiana"https://zbmath.org/authors/?q=ai:shulman.tatianaSummary: It is an old and challenging topic to investigate for which discrete groups \(G\) the full group \(\mathrm{C}^*\)-algebra \(C^\ast(G)\) is residually finite-dimensional (RFD). In particular not much is known about how the RFD property behaves under fundamental constructions, such as amalgamated free products and HNN-extensions. In [\textit{K.~Courtney} and \textit{T.~Shulman}, Proc. Am. Math. Soc. 148, No.~2, 765--776 (2020; Zbl 1445.46043)] it was proved that central amalgamated free products of virtually abelian groups are RFD. In this paper we prove that this holds much beyond this case. Our method is based on showing a certain approximation property for characters induced from central subgroups. In particular it allows us to prove that free products of polycyclic-by-finite groups amalgamated over finitely generated central subgroups are RFD. On the other hand we prove that the class of RFD \(\mathrm{C}^*\)-algebras (and groups) is not closed under central amalgamated free products. Namely we give an example of RFD groups (in fact finitely generated amenable RF groups) whose central amalgamated free product is not RFD, moreover it is not even maximally almost periodic. This answers a question of
\textit{M.~S. Khan} and \textit{S.~A. Morris} [Trans. Am. Math. Soc. 273, 405--416, 417--432 (1982; Zbl 0496.22004)].Spectra of elements in operator space tensor products of \(\mathrm{C}^{\ast}\)-algebrashttps://zbmath.org/1491.460532022-09-13T20:28:31.338867Z"Antony, Janson"https://zbmath.org/authors/?q=ai:antony.janson"Kumar, Ajay"https://zbmath.org/authors/?q=ai:kumar.ajayThe study of various tensor products in the category of operator spaces and \(C^*\)-algebras has been an essential part of non-commutative functional analysis. Projective tensor product, the Haagerup tensor product and the Schur tensor product are few of the most important ones which have attracted the attention of many working in the area. Recently, Defant and Wiesner introduced a generalized tensor product that includes all the three tensor products mentioned above, and called this construction the \(\lambda\)-tensor product [\textit{A.~Defant} and \textit{D.~Wiesner}, J. Funct. Anal. 266, No.~9, 5493--5525 (2014; Zbl 1305.46040)].
In the present paper, the authors discuss necessary conditions for the injectivity of the canonical mapping \(i_{\lambda}\) from the \(\lambda\)-tensor product \(E \otimes_{\lambda} F\) into the operator space injective tensor product \(E \otimes_{\min} F\). Let \(A\) and \(B\) be two \(C^*\)-algebras, then \(A \otimes_{\lambda} B\) and \(A \otimes_{\min} B\) respectively denote the (completed) \(\lambda\)-tensor product and the minimal \(C^*\)-tensor product of the two. The authors examine the spectra of elementary tensors and a problem related to characterizing elements using the spectra of some members of the algebra. The results are proved in terms of Property-\(P\), which the authors introduce as: A Banach algebra \(A\) is said to have Property-\(P\) if for \(u, v \in A\), \(\sigma (ux) \cup \{0\} = \sigma(vx) \cup \{0\}\) for every \(x \in A\) if and only if \(u = v\), where \(\sigma(x)\) denotes the spectrum of an element \(x \in A\).
Reviewer: Preeti Luthra (Delhi)Improved quantum hypercontractivity inequality for the qubit depolarizing channelhttps://zbmath.org/1491.460582022-09-13T20:28:31.338867Z"Beigi, Salman"https://zbmath.org/authors/?q=ai:beigi.salmanSummary: The hypercontractivity inequality for the qubit depolarizing channel \(\Psi_t\) states that \(\|\Psi_t^{\otimes n}(X) \|_p \leq \| X \|_q\), provided that \(p \geq q > 1\) and \(t \geq \ln \sqrt{\frac{p - 1}{q - 1}} \). In this paper, we present an improvement of this inequality. We first prove an improved quantum logarithmic-Sobolev inequality and then use the well-known equivalence of logarithmic-Sobolev inequalities and hypercontractivity inequalities to obtain our main result. As applications of these results, we present an asymptotically tight quantum Faber-Krahn inequality on the hypercube and a new quantum Schwartz-Zippel lemma.
{\copyright 2021 American Institute of Physics}Constrained characteristic functions, multivariable interpolation, and invariant subspaceshttps://zbmath.org/1491.460602022-09-13T20:28:31.338867Z"Hu, Jian"https://zbmath.org/authors/?q=ai:hu.jian"Wang, Maofa"https://zbmath.org/authors/?q=ai:wang.maofa"Wang, Wei"https://zbmath.org/authors/?q=au:Wang, WeiSummary: In this paper, we present a functional model theorem for completely non-coisometric \(n\)-tuples of operators in the noncommutative variety \(\mathcal{V}_{f,\varphi,\mathcal{I}}(\mathcal{H})\) in terms of constrained characteristic functions. As an application, we prove that the constrained characteristic function is a complete unitary invariant for this class of elements, which can be viewed as the noncommutative analogue of the classical Sz.-Nagy-Foiaş functional model for completely nonunitary contractions. On the other hand, we provide a Sarason-type commutant lifting theorem. Applying this result, we solve the Nevanlinna-Pick-type interpolation problem in our setting. Moreover, we also obtain a Beurling-type characterization of the joint invariant subspaces under the operators \(B_1,\dots,B_n \), where the \(n\)-tuple \((B_1,\dots,B_n)\) is the universal model associated with the abstract noncommutative variety \(\mathcal{V}_{f,\varphi,\mathcal I}\).Perturbation analysis of the algebraic metric generalized inverse in \(L^p(\Omega,\mu)\)https://zbmath.org/1491.470012022-09-13T20:28:31.338867Z"Cao, Jianbing"https://zbmath.org/authors/?q=ai:cao.jianbing"Xue, Yifeng"https://zbmath.org/authors/?q=ai:xue.yifengSummary: Let \(X=L^p(\Omega,\mu)\) \((1<p<\infty)\) and \(T,\delta T:X\to X\) be bounded linear operators. Put \(\bar{T}=T+\delta T\). In this paper, using the notion of quasi-additivity and the concept of stable perturbation, we will give some estimates of the upper bound of \(\|\bar{T}^M-T^M\|\) in terms of the gap function. As an application of main results, we also investigate the best approximate solution problem of ill-posed operator equation.On the approximate point spectra of \(m\)-complex symmetric operators, \([m,C]\)-symmetric operators and othershttps://zbmath.org/1491.470022022-09-13T20:28:31.338867Z"Chō, Muneo"https://zbmath.org/authors/?q=ai:cho.muneo"Lee, Ji Eun"https://zbmath.org/authors/?q=ai:lee.jieun.1"Načevska Nastovska, Biljana"https://zbmath.org/authors/?q=ai:nacevska-nastovska.biljana"Saito, Taiga"https://zbmath.org/authors/?q=ai:saito.taigaSummary: In this paper we study properties of approximate point spectra of \(m\)-complex symmetric operators, \([m,C]\)-symmetric operators, and others on a complex Hilbert space \(\mathcal{H}\).An extension of several essential numerical radius inequalities of \(2\times 2\) off-diagonal operator matriceshttps://zbmath.org/1491.470032022-09-13T20:28:31.338867Z"Al-Dolat, Mohammed"https://zbmath.org/authors/?q=ai:al-dolat.mohammed"Jaradat, Imad"https://zbmath.org/authors/?q=ai:jaradat.imad"Baleanu, Dumitru"https://zbmath.org/authors/?q=ai:baleanu.dumitru-iSummary: In this work, we provide upper and lower bounds for the numerical radius of an \(n\times n\) off-diagonal operator matrix, which extends some results by \textit{A. Abu-Omar} and \textit{F. Kittaneh} [Stud. Math. 216, No. 1, 69--75 (2013; Zbl 1279.47015); Linear Algebra Appl. 468, 18--26 (2015; Zbl 1316.47005); Rocky Mt. J. Math. 45, No. 4, 1055--1064 (2015; Zbl 1339.47007)], and \textit{K. Paul} and \textit{S. Bag} [Appl. Math. Comput. 222, 231--243 (2013; Zbl 1339.30003)].Refinements of numerical radius inequalities using the Kantorovich ratiohttps://zbmath.org/1491.470042022-09-13T20:28:31.338867Z"Nikzat, Elham"https://zbmath.org/authors/?q=ai:nikzat.elham"Omidvar, Mohsen Erfanian"https://zbmath.org/authors/?q=ai:omidvar.mohsen-erfanianSummary: In this paper, we improve some numerical radius inequalities for Hilbert space operators under suitable condition. We also compare our results with some known results. As application of our result, we obtain an operator inequality.Bounds for the numerical radius of \(3\times 3\) operator matriceshttps://zbmath.org/1491.470052022-09-13T20:28:31.338867Z"Paul, Kallol"https://zbmath.org/authors/?q=ai:paul.kallol"Dastidar, Shyamsree Ghosh"https://zbmath.org/authors/?q=ai:dastidar.shyamsree-ghosh"Bag, Santanu"https://zbmath.org/authors/?q=ai:bag.santanuSummary: Operator matrices play an important role in operator inequalities. The matrix representation of an operator is always an easier way to look the properties of an operator more closely. Here we obtain bounds for the numerical radius of certain \(3\times 3\) operator matrices not unitarily equivalent to off-diagonal part of \(3\times 3\) operator matrices. Then using those results we establish some upper bounds for the numerical radius of general \(3\times 3\) operator matrices.Geometric Arveson-Douglas conjecture and holomorphic extensionhttps://zbmath.org/1491.470062022-09-13T20:28:31.338867Z"Douglas, Ronald George"https://zbmath.org/authors/?q=ai:douglas.ronald-george"Wang, Yi"https://zbmath.org/authors/?q=ai:wang.yi.7|wang.yi.9|wang.yi.5|wang.yi.8|wang.yi.10|wang.yi.4|wang.yi.6|wang.yi.3|wang.yi.1Summary: In this paper, we introduce techniques from complex harmonic analysis to prove a weaker version of the geometric Arveson-Douglas conjecture on the Bergman space for a complex analytic subset that is smooth on the boundary of the unit ball and intersects transversally with it. In fact, we prove that the projection operator onto the corresponding quotient module is in the Toeplitz algebra \(\mathcal{T}(L^{\infty})\), which implies the essential normality of the quotient module. Combining some other techniques, we actually obtain the \(p\)-essential normality for \(p>2d\), where \(d\) is the complex dimension of the analytic subset. Finally, we show that our results apply to the closure of a radical polynomial ideal \(I\) whose zero variety satisfies the above conditions. A~key technique is defining a right inverse operator of the restriction map from the unit ball to the analytic subset, generalizing the result of \textit{F. Beatrous jun.}'s paper [Mich. Math. J. 32, 361--380 (1985; Zbl 0584.32024)].Essential normality -- a unified approach in terms of local decompositionshttps://zbmath.org/1491.470072022-09-13T20:28:31.338867Z"Wang, Yi"https://zbmath.org/authors/?q=ai:wang.yi.3|wang.yi.7|wang.yi.6|wang.yi.8|wang.yi.4|wang.yi.9|wang.yi.5|wang.yi.1|wang.yi.10|wang.yi.2Summary: In this paper, we define the asymptotic stable division property for submodules of \(L_a ^2 (\mathbb{B}_n)\). We show that under a mild condition, a~submodule with the asymptotic stable division property is \(p\)-essentially normal for all \(p > n\). A~new technique is developed to show that certain submodules have the asymptotic stable division property. This leads to a unified proof of most known results on essential normality of submodules as well as new results. In particular, we show that an ideal defines a \(p\)-essentially normal submodule of \(L_a ^2 (\mathbb B_n )\), \( \forall p > n\), if its associated primary ideals are powers of prime ideals whose zero loci satisfy standard regularity conditions near the sphere.Improved inequalities for the numerical radius: when inverse commutes with the normhttps://zbmath.org/1491.470082022-09-13T20:28:31.338867Z"Cain, Bryan E."https://zbmath.org/authors/?q=ai:cain.bryan-eSummary: New inequalities relating the norm \(n(X)\) and the numerical radius \(w(X)\) of invertible bounded linear Hilbert space operators were announced by \textit{M. S. Hosseini} and \textit{M. E. Omidvar} [Bull. Aust. Math. Soc. 94, No. 3, 489--496 (2016; Zbl 1387.47005)]. For example, they asserted that \(w(AB)\leq 2w(A)w(B)\) for invertible bounded linear Hilbert space operators \(A\) and \(B\). We identify implicit hypotheses used in their discovery. The inequalities and their proofs can be made good by adding the extra hypotheses which take the form \(n(X^{-1})=n(X)^{-1}\). We give counterexamples in the absence of such additional hypotheses. Finally, we show that these hypotheses yield even stronger conclusions, for example, \(w(AB)=w(A)w(B)\).A Beurling-Lax-Halmos theorem for spaces with a complete Nevanlinna-Pick factorhttps://zbmath.org/1491.470092022-09-13T20:28:31.338867Z"Clouâtre, Raphaël"https://zbmath.org/authors/?q=ai:clouatre.raphael"Hartz, Michael"https://zbmath.org/authors/?q=ai:hartz.michael"Schillo, Dominik"https://zbmath.org/authors/?q=ai:schillo.dominikSummary: We provide a short argument to establish a Beurling-Lax-Halmos theorem for reproducing kernel Hilbert spaces whose kernel has a complete Nevanlinna-Pick factor. We also record factorization results for pairs of nested invariant subspaces.Stability criteria for positive linear discrete-time systemshttps://zbmath.org/1491.470102022-09-13T20:28:31.338867Z"Glück, Jochen"https://zbmath.org/authors/?q=ai:gluck.jochen"Mironchenko, Andrii"https://zbmath.org/authors/?q=ai:mironchenko.andriiSummary: We prove new characterisations of exponential stability for positive linear discrete-time systems in ordered Banach spaces, in terms of small-gain conditions. Such conditions have played an important role in the finite-dimensional systems theory, but are relatively unexplored in the infinite-dimensional setting, yet. Our results are applicable to discrete-time systems in ordered Banach spaces that have a normal and generating positive cone. Moreover, we show that our stability criteria can be considerably simplified if the cone has non-empty interior or if the operator under consideration is quasi-compact. To place our results into context we include an overview of known stability criteria for linear (and not necessarily positive) operators and provide full proofs for several folklore characterizations from this domain.Perturbation bounds for eigenspaces under a relative gap conditionhttps://zbmath.org/1491.470112022-09-13T20:28:31.338867Z"Jirak, Moritz"https://zbmath.org/authors/?q=ai:jirak.moritz"Wahl, Martin"https://zbmath.org/authors/?q=ai:wahl.martinSummary: A basic problem in operator theory is to estimate how a small perturbation affects the eigenspaces of a self-adjoint compact operator. In this paper, we prove upper bounds for the subspace distance, tailored for relative perturbations. As a main example, we consider the empirical covariance operator and show that a sharp bound can be achieved under a relative gap condition. The proof is based on a novel contraction phenomenon, contrasting previous spectral perturbation approaches.Clark model in the general situationhttps://zbmath.org/1491.470122022-09-13T20:28:31.338867Z"Liaw, Constanze"https://zbmath.org/authors/?q=ai:liaw.constanze"Treil, Sergei"https://zbmath.org/authors/?q=ai:treil.sergeiSummary: For a unitary operator \(U\) in a Hilbert space \(\mathcal H\), the family of its unitary perturbations by rank one operators with fixed range is parametrized by a complex parameter \(\gamma\), \(|\gamma| = 1\). Namely, all such unitary perturbations are operators \(U_{\gamma}:= U + (\gamma - 1)(\cdot, b_{1})_{\mathcal H}b\), where \(b\in \mathcal H\), \(\|b\| = 1\), \(b_{1} = U^{-1}b\), \(|\gamma| = 1\). For \(|\gamma| < 1\), the operators \(U_{\gamma}\) are contractions with one-dimensional defects.
Restricting our attention to the non-trivial part of perturbation, we assume that \(b\) is a cyclic vector for \(U\), i.e., \({\mathcal H} = \overline {\mathrm{span}} \{U^nb:n \in \mathbb{Z} \}\). In this case, the operator \(U_{\gamma}\), \(|\gamma| < 1\) is a completely non-unitary contraction and thus unitarily equivalent to its functional model \(\mathcal{M}_{\gamma}\), which is the compression of the multiplication by the independent variable \(z\) onto the model space \({\mathcal{K}_{{\theta _\gamma }}}\); here, \(\theta_{\gamma}\) is the characteristic function of the contraction \(U_{\gamma}\).
The Clark operator \(\Phi_{\gamma}\) is a unitary operator intertwining the operator \(U_{\gamma}, |\gamma| < 1\) (in the spectral representation of the operator \(U\)) and its model \(\mathcal{M}_{\gamma}\), \(\mathcal{M}_{\gamma}\Phi_{\gamma} = \Phi_{\gamma}U_{\gamma}\). In the case when the spectral measure of \(U\) is purely singular (equivalently, the characteristic function \(\theta_{\gamma}\) is inner), the operator \(\Phi_{\gamma}\) was described from a slightly different point of view by \textit{D. N. Clark} [J. Anal. Math. 25, 169--191 (1972; Zbl 0252.47010)]. The case where \(\theta_{\gamma}\) is an extreme point of the unit ball in \(H^{\infty}\) was treated by \textit{D. Sarason} [Sub-Hardy Hilbert spaces in the unit disk. New York, NY: John Wiley \& Sons (1994; Zbl 1253.30002)], using the sub-Hardy spaces \(\mathcal H(\theta)\) introduced by {L.\,de\,Branges}.
In this paper, we treat the general case and give a systematic presentation of the subject. We first find a formula for the adjoint operator \(\Phi_{\gamma}^{*}\), which is represented by a~singular integral operator, generalizing in a sense the normalized Cauchy transform studied by \textit{A. G. Poltoratskij} [St. Petersbg. Math. J. 5, No. 2, 1 (1993; Zbl 0833.30018); translation from Algebra Anal. 5, No. 2, 189--210 (1993)]. We begin by presenting a ``universal'' representation that works for any transcription of the functional model. We then give the formulas adapted for specific transcriptions of the model, such as Sz.-Nagy-Foiaş and the de Branges-Rovnyak transcriptions. Finally, we obtain the representation of \(\Phi_{\gamma}\).A new type of operator convexityhttps://zbmath.org/1491.470132022-09-13T20:28:31.338867Z"Dinh, Trung-Hoa"https://zbmath.org/authors/?q=ai:dinh-trung-hoa."Dinh, Thanh-Duc"https://zbmath.org/authors/?q=ai:dinh.thanh-duc"Vo, Bich-Khue T."https://zbmath.org/authors/?q=ai:vo.bich-khue-tSummary: Let \(r, s\) be positive numbers. We define a new class of operator \((r, s)\)-convex functions by the following inequality
\[
f \left( \left[\lambda A^{r} + (1-\lambda)B^{r}\right]^{1/r}\right) \leq [\lambda f(A)^{s} +(1-\lambda)f(B)^{s}]^{1/s},
\]
where \(A, B\) are positive definite matrices and for any \(\lambda \in [0,1]\). We prove the Jensen, Hansen-Pedersen, and Rado type inequalities for such functions. Some equivalent conditions for a function \(f\) to become operator \((r, s)\)-convex are established.Inequalities for relative operator entropies and operator meanshttps://zbmath.org/1491.470142022-09-13T20:28:31.338867Z"Furuichi, Shigeru"https://zbmath.org/authors/?q=ai:furuichi.shigeru"Minculete, Nicuşor"https://zbmath.org/authors/?q=ai:minculete.nicusorSummary: The main purpose of this article is to study estimates for the Tsallis relative operator entropy, by using the Hermite-Hadamard inequality. We obtain alternative bounds for the Tsallis relative operator entropy and in the process to derive these bounds, we established the significant relation between the Tsallis relative operator entropy and the generalized relative operator entropy. In addition, we study the properties on monotonicity for the weight of operator means and for the parameter of relative operator entropies.Furuta type inequalities via operator means and applications to Kadison's type inequalitieshttps://zbmath.org/1491.470152022-09-13T20:28:31.338867Z"Matharu, Jagjit Singh"https://zbmath.org/authors/?q=ai:matharu.jagjit-singh"Yamazaki, Takeaki"https://zbmath.org/authors/?q=ai:yamazaki.takeaki"Malhotra, Chitra"https://zbmath.org/authors/?q=ai:malhotra.chitraSummary: We give a characterization of chaotic order via an arbitrary operator mean \(\sigma\) as follows. For \(p,r > 0\),
\[
\begin{aligned} \log A \geq \log B \quad \text{if and only if} \quad A^{-r\alpha} \sigma_h B^{p\alpha} \leq I, \end{aligned}
\]
for all \(\alpha \geq 0\), where \(A\) and \(B\) are positive invertible operators, \(h\) is a normalized operator monotone function on \((0,\infty)\) satisfying \(h(t^s)\leq h(t)^s\) for all \(t>0, s \geq 1\) and \(h'(1)=\frac{r}{p+r}\). It is a generalization of the well-known characterization of chaotic order using operator geometric mean. We also obtain Furuta-type inequalities via operator means. As applications of the result, we generalize an asymmetric Kadison's inequality as follows:
\[
\begin{aligned} h_{\alpha}\left( \left| \phi (A^p)^{\lambda} \phi (A^q)^{\mu } \right|^2 \right) \leq \phi (A^{2\alpha (p\lambda +q\mu)}) \end{aligned}
\]
for all \(p,q,\lambda, \mu \geq 0\) satisfying \(2\alpha (p\lambda +q\mu)\leq p+2q\mu\), \(q\leq 2\alpha (p\lambda +q\mu)\leq 2q\), \(0\leq p\leq q\) and unital positive linear map \(\phi\).Around Jensen's inequality for strongly convex functionshttps://zbmath.org/1491.470162022-09-13T20:28:31.338867Z"Moradi, Hamid Reza"https://zbmath.org/authors/?q=ai:moradi.hamid-reza"Omidvar, Mohsen Erfanian"https://zbmath.org/authors/?q=ai:omidvar.mohsen-erfanian"Khan, Muhammad Adil"https://zbmath.org/authors/?q=ai:khan.muhammad-adil"Nikodem, Kazimierz"https://zbmath.org/authors/?q=ai:nikodem.kazimierzSummary: In this paper, we use basic properties of strongly convex functions to obtain new inequalities including Jensen type and Jensen-Mercer type inequalities. Applications for special means are pointed out as well. We also give a Jensen's operator inequality for strongly convex functions. As a corollary, we improve the Hölder-McCarthy inequality under suitable conditions. More precisely we show that if \(Sp(A) \subset (1,\infty)\), then
\[
\langle Ax,x \rangle^r\leq \langle A^rx,x \rangle -\frac{r^2-r}{2}\left(\langle A^2x,x\rangle -\langle Ax,x \rangle ^2 \right),\quad r\geq 2
\]
and if \(Sp(A) \subset (0,1)\), then
\[
\langle A^rx,x\rangle \leq\langle Ax,x \rangle^r+\frac{r-r^2}{2}\left(\langle Ax,x \rangle^2-\langle A^2x,x\rangle \right),\quad 0<r<1
\]
for each positive operator \(A\) and \(x\in \mathcal {H}\) with \(\| x\| =1\).Operator functions and the operator harmonic meanhttps://zbmath.org/1491.470172022-09-13T20:28:31.338867Z"Uchiyama, Mitsuru"https://zbmath.org/authors/?q=ai:uchiyama.mitsuruSummary: The objective of this paper is to investigate operator functions by making use of the operator harmonic mean `!'. For \(0<A\leqq B\), we construct a unique pair \(X, Y\) such that \(0<X\leqq Y\), \(A=X\operatorname{!}Y\), \(B=\frac{X+Y}{2} \). We next give a condition for operators \(A, B, C\geqq 0\) in order that \(C \leqq A\operatorname{!}B\) and show that \(g\ne 0\) is strongly operator convex on \(J\) if and only if \(g(t)>0\) and \(g (\frac{A+B}{2}) \leqq g(A)\operatorname{!}g(B)\) for \(A, B\) with spectra in \(J\). This inequality particularly holds for an operator decreasing function on the right half line. We also show that \(f(t)\) defined on \((0, b)\) with \(0<b\leqq \infty\) is operator monotone if and only if \(f(0+)<\infty\), \(f (A\operatorname{!}B)\leqq \frac{1}{2}(f(A) + f(B))\). In particular, if \(f>0\), then \(f\) is operator monotone if and only if \(f (A\operatorname{!}B) \leqq f(A)\operatorname{!}f(B)\). We lastly prove that if a strongly operator convex function \(g(t)>0\) on a finite interval \((a, b)\) is operator decreasing, then \(g\) has an extension \(\tilde{g}\) to \((a, \infty )\) that is positive and operator decreasing.Projections in the convex hull of isometries on \(C^2 [0,1]\)https://zbmath.org/1491.470182022-09-13T20:28:31.338867Z"Baker, Abdullah Bin Abu"https://zbmath.org/authors/?q=ai:baker.abdullah-bin-abu"Maurya, Rahul"https://zbmath.org/authors/?q=ai:maurya.rahul-kumarSummary: Let \(C^2 [0, 1]\) be the Banach space of all functions that have continuous derivatives \(f^{\prime}\) and \(f^{\prime\prime}\) on the closed interval \([0, 1]\), equipped with norm \(\Vert f\Vert = |f(0)| + |f^{\prime}(0)| + \Vert f^{\prime\prime}\Vert_{\infty}\), where \(\Vert \cdot \Vert_{\infty}\) is the usual supremum norm. In this paper, we characterize projections on \(C^2 [0, 1]\) that can be written as convex combination of two surjective linear isometries. We also find out the structure of Hermitian projections and generalized bi-circular projections on \(C^2 [0, 1]\). Finally, we discuss the relationship of these two types of projections (Hermitian and generalized bi-circular projections) with the convex combination of two isometries.Spectrum of weighted composition operators. V: Spectrum and essential spectra of weighted rotation-like operatorshttps://zbmath.org/1491.470192022-09-13T20:28:31.338867Z"Kitover, Arkady"https://zbmath.org/authors/?q=ai:kitover.arkady-k"Orhon, Mehmet"https://zbmath.org/authors/?q=ai:orhon.mehmetIn this paper, the notion of a ``rotation-like'' operator is presented. The spectrum and essential spectra of these operators are computed in detail. Then these operators are studied on some particular cases, such as on Banach ideal spaces, spaces of analytic functions, and on uniform algebras, respectively. Finally, the authors apply some results to investigate the essential spectra of weighted ``rotation-like'' operators in various Banach spaces of analytic functions.
For Parts I--IV, see the authors' papers [\textit{A. K. Kitover}, Positivity 15, No. 4, 639--659 (2011; Zbl 1284.47023); ibid. 17, No. 3, 655--676 (2013; Zbl 1288.47027); in: Ordered structures and applications. Positivity VII. Basel: Birkhäuser/Springer. 233--261 (2016; Zbl 1411.47007); Positivity 21, No. 3, 989--1014 (2017; Zbl 1439.47023)], respectively.
Reviewer: Pham Viet Hai (Hanoi)Ambarzumian-type problems for discrete Schrödinger operatorshttps://zbmath.org/1491.470232022-09-13T20:28:31.338867Z"Hatinoğlu, Burak"https://zbmath.org/authors/?q=ai:hatinoglu.burak"Eakins, Jerik"https://zbmath.org/authors/?q=ai:eakins.jerik"Frendreiss, William"https://zbmath.org/authors/?q=ai:frendreiss.william"Lamb, Lucille"https://zbmath.org/authors/?q=ai:lamb.lucille"Manage, Sithija"https://zbmath.org/authors/?q=ai:manage.sithija"Puente, Alejandra"https://zbmath.org/authors/?q=ai:puente.alejandraSummary: We discuss the problem of unique determination of the finite free discrete Schrödinger operator from its spectrum, also known as the Ambarzumian problem [\textit{V. Ambarzumian}, Z. Phys. 53, 690--695 (1929; JFM 55.0868.01)], with various boundary conditions, namely, any real constant boundary condition at zero and Floquet boundary conditions of any angle. Then we prove the following Ambarzumian-type mixed inverse spectral problem: diagonal entries except the first and second ones and a set of two consecutive eigenvalues uniquely determine the finite free discrete Schrödinger operator.Approximations of self-adjoint \(C_0\)-semigroups in the operator-norm topologyhttps://zbmath.org/1491.470332022-09-13T20:28:31.338867Z"Zagrebnov, Valentin"https://zbmath.org/authors/?q=ai:zagrebnov.valentin-aSummary: The paper improves approximation theory based on the Trotter-Kato product formulae. For self-adjoint \(C_0\)-semigroups we develop a lifting of the strongly convergent Chernoff approximation (or \textit{product}) formula to convergence in the operator-norm topology. This allows to obtain optimal estimate for the rate of operator-norm convergence of Trotter-Kato product formulae for Kato functions from the class \(K_2\).Second-order differential operators in the limit circle casehttps://zbmath.org/1491.470352022-09-13T20:28:31.338867Z"Yafaev, Dmitri R."https://zbmath.org/authors/?q=ai:yafaev.dimitri-rSummary: We consider symmetric second-order differential operators with real coefficients such that the corresponding differential equation is in the limit circle case at infinity. Our goal is to construct the theory of self-adjoint realizations of such operators by an analogy with the case of Jacobi operators. We introduce a new object, the quasiresolvent of the maximal operator, and use it to obtain a very explicit formula for the resolvents of all self-adjoint realizations. In particular, this yields a simple representation for the Cauchy-Stieltjes transforms of the spectral measures playing the role of the classical Nevanlinna formula in the theory of Jacobi operators.Symbolic computation applied to the study of the kernel of special classes of paired singular integral operatorshttps://zbmath.org/1491.682752022-09-13T20:28:31.338867Z"Conceição, Ana C."https://zbmath.org/authors/?q=ai:conceicao.ana-cSummary: Operator theory has many applications in several main scientific research areas (structural mechanics, aeronautics, quantum mechanics, ecology, probability theory, electrical engineering, among others) and the importance of its study is globally acknowledged. On the study of the operator's kernel some progress has been achieved for some specific classes of singular integral operators whose properties allow the use of particular strategies. However, the existing algorithms allow, in general, to study the dimension of the kernel of some classes of singular integral operators but are not designed to be implemented on a computer. The main goal of this paper is to show how the symbolic and numeric capabilities of a computer algebra system can be used to study the kernel of special classes of paired singular integral operators with essentially bounded coefficients defined on the unit circle. It is described how some factorization algorithms can be used to compute the dimension of the kernel of special classes of singular integral operators. The analytical algorithms [ADimKerPaired-Scalar], [AKerPaired-Scalar], and [ADimKerPaired-Matrix] are presented. The design of these new algorithms was focused on the possibility of implementing on a computer all the extensive symbolic and numeric calculations present in the algorithms. For the essentially bounded hermitian coefficients case, there exist some relations with Hankel operators. The paper contains some interesting and nontrivial examples obtained with the use of a computer algebra system.Two-particle bound states at interfaces and cornershttps://zbmath.org/1491.810182022-09-13T20:28:31.338867Z"Roos, Barbara"https://zbmath.org/authors/?q=ai:roos.barbara"Seiringer, Robert"https://zbmath.org/authors/?q=ai:seiringer.robertSummary: We study two interacting quantum particles forming a bound state in \(d\)-dimensional free space, and constrain the particles in \(k\) directions to \(( 0 , \infty )^k \times \mathbb{R}^{d - k} \), with Neumann boundary conditions. First, we prove that the ground state energy strictly decreases upon going from \(k\) to \(k + 1\). This shows that the particles stick to the corner where all boundary planes intersect. Second, we show that for all \(k\) the resulting Hamiltonian, after removing the free part of the kinetic energy, has only finitely many eigenvalues below the essential spectrum. This paper generalizes the work of \textit{S. Egger} et al. [J. Spectr. Theory 10, No. 4, 1413--1444 (2020; Zbl 1469.81021)] to dimensions \(d > 1\).Constraining light mediators via detection of coherent elastic solar neutrino nucleus scatteringhttps://zbmath.org/1491.810282022-09-13T20:28:31.338867Z"Li, Yu-Feng"https://zbmath.org/authors/?q=ai:li.yufeng|li.yu-feng"Xia, Shuo-yu"https://zbmath.org/authors/?q=ai:xia.shuo-yuSummary: Dark matter (DM) direct detection experiments are entering the multiple-ton era and will be sensitive to the coherent elastic neutrino nucleus scattering (CE\(\nu\)NS) of solar neutrinos, enabling the possibility to explore contributions from new physics with light mediators at the low energy range. In this paper we consider light mediator models (scalar, vector and axial vector) and the corresponding contributions to the solar neutrino CE\(\nu\)NS process. Motivated by the current status of new generation of DM direct detection experiments and the future plan, we study the sensitivity of light mediators in DM direct detection experiments of different nuclear targets and detector techniques. The constraints from the latest \(^8\mathrm{B}\) solar neutrino measurements of XENON-1T are also derived. Finally, we show that the solar neutrino CE\(\nu\)NS process can provide stringent limitation on the \(L_\mu - L_\tau\) model with the vector mediator mass below 100 MeV, covering the viable parameter space of the solution to the \((g - 2)_\mu\) anomaly.The 28 GeV dimuon excess in lepton specific THDMhttps://zbmath.org/1491.810302022-09-13T20:28:31.338867Z"Çiçi, Ali"https://zbmath.org/authors/?q=ai:cici.ali"Khalil, Shaaban"https://zbmath.org/authors/?q=ai:khalil.shaaban"Niş, Büşra"https://zbmath.org/authors/?q=ai:nis.busra"Ün, Cem Salih"https://zbmath.org/authors/?q=ai:un.cem-salihSummary: We explore the Higgs mass spectrum in a class of Two Higgs Doublet Models (THDMs) in which a scalar \(SU(2)_L\) doublet interacts only with quarks, while the second one interacts only with leptons. The spectrum includes two CP-even Higgs bosons, either of which can account for the SM-like Higgs boson, and the spectra involving light Higgs bosons receive strong impacts from the LEP results and the current collider analyses. We find that a consistent spectrum can involve a CP-odd Higgs boson as light as about 10 GeV, while the lightest CP-even Higgs boson cannot be lighter than about 55 GeV when \(m_A \sim 28\) GeV. These analyses can rather bound the low \(\tan\beta\) region which can also accommodate an observed excess in dimuon events at \(m_{\mu\mu} \sim 28\) GeV. A lepton-specific class of THDMs (LS-THDM) can predict such an excess through \(A \to \mu\mu\) decays, while the solutions can be constrained by the \(A \to \tau\tau\) mode. After constraining the solutions with the consistent ranges of \(\sigma(pp \to bbA \to bb\tau\tau)\), a largest excess at about \(1.5 \sigma\) at 8 TeV center of mass (COM) energy and \(2\sigma\) at 13 TeV COM is observed for \(\tan\beta \sim 12\) and \(m_A \sim 28\) GeV in the \(\sigma(pp \to bbA \to bb\mu\mu)\) events.Relativistic and non-Gaussianity contributions to the one-loop power spectrumhttps://zbmath.org/1491.830052022-09-13T20:28:31.338867Z"Martinez-Carrillo, Rebeca"https://zbmath.org/authors/?q=ai:martinez-carrillo.rebeca"De-Santiago, Josue"https://zbmath.org/authors/?q=ai:de-santiago.josue"Hidalgo, Juan Carlos"https://zbmath.org/authors/?q=ai:hidalgo.juan-carlos"Malik, Karim A."https://zbmath.org/authors/?q=ai:malik.karim-a(no abstract)On the expected backreaction during preheatinghttps://zbmath.org/1491.830062022-09-13T20:28:31.338867Z"Armendariz-Picon, C."https://zbmath.org/authors/?q=ai:armendariz-picon.cristian(no abstract)Probing multi-step electroweak phase transition with multi-peaked primordial gravitational waves spectrahttps://zbmath.org/1491.830132022-09-13T20:28:31.338867Z"Morais, António P."https://zbmath.org/authors/?q=ai:morais.antonio-p"Pasechnik, Roman"https://zbmath.org/authors/?q=ai:pasechnik.roman(no abstract)Beyond limber: efficient computation of angular power spectra for galaxy clustering and weak lensinghttps://zbmath.org/1491.830202022-09-13T20:28:31.338867Z"Fang, Xiao"https://zbmath.org/authors/?q=ai:fang.xiao.1|fang.xiao"Krause, Elisabeth"https://zbmath.org/authors/?q=ai:krause.elisabeth"Eifler, Tim"https://zbmath.org/authors/?q=ai:eifler.tim"MacCrann, Niall"https://zbmath.org/authors/?q=ai:maccrann.niall(no abstract)Parametrising non-linear dark energy perturbationshttps://zbmath.org/1491.830222022-09-13T20:28:31.338867Z"Hassani, Farbod"https://zbmath.org/authors/?q=ai:hassani.farbod"L'Huillier, Benjamin"https://zbmath.org/authors/?q=ai:lhuillier.benjamin"Shafieloo, Arman"https://zbmath.org/authors/?q=ai:shafieloo.arman"Kunz, Martin"https://zbmath.org/authors/?q=ai:kunz.martin"Adamek, Julian"https://zbmath.org/authors/?q=ai:adamek.julian(no abstract)Buchdahl compactness limit and gravitational field energyhttps://zbmath.org/1491.830312022-09-13T20:28:31.338867Z"Dadhich, Naresh"https://zbmath.org/authors/?q=ai:dadhich.naresh(no abstract)Spiky CMB distortions from primordial bubbleshttps://zbmath.org/1491.830322022-09-13T20:28:31.338867Z"Deng, Heling"https://zbmath.org/authors/?q=ai:deng.heling(no abstract)Primordial black holes from a tiny bump/dip in the inflaton potentialhttps://zbmath.org/1491.830352022-09-13T20:28:31.338867Z"Mishra, Swagat S."https://zbmath.org/authors/?q=ai:mishra.swagat-s"Sahni, Varun"https://zbmath.org/authors/?q=ai:sahni.varun(no abstract)Spontaneous radiation of black holeshttps://zbmath.org/1491.830362022-09-13T20:28:31.338867Z"Zeng, Ding-fang"https://zbmath.org/authors/?q=ai:zeng.ding-fangSummary: We provide an explicitly hermitian hamiltonian description for the spontaneous radiation of black holes, which is a many-level, multiple-degeneracy generalization of the usual Janeys-Cummings model for two-level atoms. We show that under single-particle radiation and standard Wigner-Wiesskopf approximation, our model yields exactly thermal type power spectrum as hawking radiation requires. While in the many-particle radiation cases, numeric methods allow us to follow the evolution of microscopic state of a black hole exactly, from which we can get the firstly increasing then decreasing entropy variation trend for the radiation particles just as the Page-curve exhibited. Basing on this model analysis, we claim that two ingredients are necessary for resolutions of the information missing puzzle, a spontaneous radiation like mechanism for the production of hawking particles and proper account of the macroscopic superposition happening in the full quantum description of a black hole radiation evolution and, the working logic of replica wormholes is an effect account of this latter ingredient.
As the basis for our interpretation of black hole Hawking radiation as their spontaneous radiation, we also provide a fully atomic like inner structure models for their microscopic states definition and origins of their Bekenstein-Hawking entropy, that is, exact solution families to the Einstein equation sourced by matter constituents oscillating across the central point and their quantization. Such a first quantization model for black holes' microscopic state is non necessary for our spontaneous radiation description, but has advantages comparing with other alternatives, such as string theory fuzzball or brick wall models.Generalized uncertainty principle effects in the Hořava-Lifshitz quantum theory of gravityhttps://zbmath.org/1491.830372022-09-13T20:28:31.338867Z"García-Compeán, H."https://zbmath.org/authors/?q=ai:garcia-compean.hugo|garcia-compean.hector"Mata-Pacheco, D."https://zbmath.org/authors/?q=ai:mata-pacheco.dSummary: The Wheeler-DeWitt equation for a Kantowski-Sachs metric in Hořava-Lifshitz gravity with a set of coordinates in minisuperspace that obey a generalized uncertainty principle is studied. We first study the equation coming from a set of coordinates that obey the usual uncertainty principle and find analytic solutions in the infrared as well as a particular ultraviolet limit that allows us to find the solution found in Hořava-Lifshitz gravity with projectability and with detailed balance but now as an approximation of the theory without detailed balance. We then consider the coordinates that obey the generalized uncertainty principle by modifying the previous equation using the relations between both sets of coordinates. We describe two possible ways of obtaining the Wheeler-DeWitt equation. One of them is useful to present the general equation but it is found to be very difficult to solve. Then we use the other proposal to study the limiting cases considered before, that is, the infrared limit that can be compared to the equation obtained by using general relativity and the particular ultraviolet limit. For the second limit we use a ultraviolet approximation and then solve analytically the resulting equation. We find that and oscillatory behaviour is possible in this context but it is not a general feature for any values of the parameters involved.Many-field inflation: universality or prior dependence?https://zbmath.org/1491.830502022-09-13T20:28:31.338867Z"Christodoulidis, Perseas"https://zbmath.org/authors/?q=ai:christodoulidis.perseas"Roest, Diederik"https://zbmath.org/authors/?q=ai:roest.diederik"Rosati, Robert"https://zbmath.org/authors/?q=ai:rosati.robert(no abstract)Testing kinetically coupled inflation models with CMB distortionshttps://zbmath.org/1491.830512022-09-13T20:28:31.338867Z"Dai, Rui"https://zbmath.org/authors/?q=ai:dai.rui"Zhu, Yi"https://zbmath.org/authors/?q=ai:zhu.yi.1|zhu.yi|zhu.yi.3|zhu.yi.2(no abstract)Pseudosmooth tribrid inflation in SU(5)https://zbmath.org/1491.830522022-09-13T20:28:31.338867Z"Masoud, Muhammad Atif"https://zbmath.org/authors/?q=ai:masoud.muhammad-atif"Rehman, Mansoor Ur"https://zbmath.org/authors/?q=ai:rehman.mansoor-ur"Shafi, Qaisar"https://zbmath.org/authors/?q=ai:shafi.qaisar(no abstract)On the slope of the curvature power spectrum in non-attractor inflationhttps://zbmath.org/1491.830532022-09-13T20:28:31.338867Z"Özsoy, Ogan"https://zbmath.org/authors/?q=ai:ozsoy.ogan"Tasinato, Gianmassimo"https://zbmath.org/authors/?q=ai:tasinato.gianmassimo(no abstract)Cosmic microwave background anisotropy numerical solution (CMBAns). I: An introduction to \(C_l\) calculationhttps://zbmath.org/1491.830592022-09-13T20:28:31.338867Z"Das, Santanu"https://zbmath.org/authors/?q=ai:das.santanu-kumar"Phan, Anh"https://zbmath.org/authors/?q=ai:phan.anh-dung|phan.anh-vu|phan.anh-huy(no abstract)Capturing non-Gaussianity of the large-scale structure with weighted skew-spectrahttps://zbmath.org/1491.830602022-09-13T20:28:31.338867Z"Dizgah, Azadeh Moradinezhad"https://zbmath.org/authors/?q=ai:dizgah.azadeh-moradinezhad"Lee, Hayden"https://zbmath.org/authors/?q=ai:lee.hayden"Schmittfull, Marcel"https://zbmath.org/authors/?q=ai:schmittfull.marcel"Dvorkin, Cora"https://zbmath.org/authors/?q=ai:dvorkin.cora(no abstract)Multipoles of the relativistic galaxy bispectrumhttps://zbmath.org/1491.850042022-09-13T20:28:31.338867Z"de Weerd, Eline M."https://zbmath.org/authors/?q=ai:de-weerd.eline-m"Clarkson, Chris"https://zbmath.org/authors/?q=ai:clarkson.chris-a"Jolicoeur, Sheean"https://zbmath.org/authors/?q=ai:jolicoeur.sheean"Maartens, Roy"https://zbmath.org/authors/?q=ai:maartens.roy"Umeh, Obinna"https://zbmath.org/authors/?q=ai:umeh.obinna(no abstract)