Recent zbMATH articles in MSC 47Bhttps://www.zbmath.org/atom/cc/47B2021-04-16T16:22:00+00:00WerkzeugAn operational construction of the sum of two non-commuting observables in quantum theory and related constructions.https://www.zbmath.org/1456.810102021-04-16T16:22:00+00:00"Drago, Nicolò"https://www.zbmath.org/authors/?q=ai:drago.nicolo"Mazzucchi, Sonia"https://www.zbmath.org/authors/?q=ai:mazzucchi.sonia"Moretti, Valter"https://www.zbmath.org/authors/?q=ai:moretti.valterSummary: The existence of a real linear space structure on the set of observables of a quantum system -- i.e., the requirement that the linear combination of two generally non-commuting observables \(A, B\) is an observable as well -- is a fundamental postulate of the quantum theory yet before introducing any structure of algebra. However, it is by no means clear how to choose the measuring instrument of a general observable of the form \(aA+bB\) (\(a,b\in\mathbb{R}\)) if such measuring instruments are given for the addends observables \(A\) and \(B\) when they are incompatible observables. A mathematical version of this dilemma is how to construct the spectral measure of \(f(aA+bB)\) out of the spectral measures of \(A\) and \(B\). We present such a construction with a formula which is valid for general unbounded self-adjoint operators \(A\) and \(B\), whose spectral measures may not commute, and a wide class of functions \(f: \mathbb{R}\rightarrow\mathbb{C} \). In the bounded case, we prove that the Jordan product of \(A\) and \(B\) (and suitably symmetrized polynomials of \(A\) and \(B)\) can be constructed with the same procedure out of the spectral measures of \(A\) and \(B\). The formula turns out to have an interesting operational interpretation and, in particular cases, a nice interplay with the theory of Feynman path integration and the Feynman-Kac formula.A basis of \(\mathbb{R}^n\) with good isometric properties and some applications to denseness of norm attaining operators.https://www.zbmath.org/1456.460102021-04-16T16:22:00+00:00"Acosta, María D."https://www.zbmath.org/authors/?q=ai:acosta.maria-d"Dávila, José L."https://www.zbmath.org/authors/?q=ai:davila.jose-lThis paper deals with the Bishop-Phelps-Bollobás property introduced in [\textit{M. D. Acosta} et al., J. Funct. Anal. 254, No. 11, 2780--2799 (2008; Zbl 1152.46006)]. The Bishop-Phelps-Bollobás property of a pair \((X,Y)\) is an improved version of the denseness of the set of norm attaining (bounded linear) operators from \(X\) to \(Y\), in the same way that \textit{B. Bollobás} [Bull. Lond. Math. Soc., 2, 181--182 (1970; Zbl 0217.45104)] improved the classical Bishop-Phelps theorem [\textit{E. Bishop} and \textit{R. R. Phelps}, Bull. Am. Math. Soc., 67, 97--98 (1961; Zbl 0098.07905)]. A surprising result contained in the seminal paper [Acosta et al., loc. cit.] is that the Bishop-Phelps-Bollobás property may fail for a pair \((X,Y)\) even when the Banach space \(X\) is finite dimensional, despite the fact that, in this case, all operators from \(X\) to \(Y\) attain their norms. In that paper, the case when the domain space \(X\) is \(\ell_1^n\) (the \(n\)-dimensional \(\ell_1\) space) or even when \(X=\ell_1\), was studied and the Bishop-Phelps-Bollobás property was characterized in terms of a property of \(Y\) called the approximate hyperplane series property (AHSP), which deals with convex series in the space \(Y\). On the other hand, it is known that when \(X\) is a uniformly convex Banach space, then every pair \((X,Y)\) has the Bishop-Phelps-Bollobás property, regardless of the range space \(Y\), see [\textit{S. K. Kim} and \textit{H. J. Lee}, Can. J. Math. 66, No. 2, 373--386 (2014; Zbl 1298.46016)] or [\textit{M. D. Acosta} et al., Trans. Am. Math. Soc. 365, No. 11, 5911--5932 (2013; Zbl 1296.46007)].
In the paper under review, the case when \(X\) is \(\ell_\infty^n\) (the \(n\)-dimensional \(\ell_\infty\) space) is studied. The authors characterize those Banach spaces \(Y\) for which the pair \((\ell_\infty^n,Y)\) has the Bishop-Phelps-Bollobás property in terms of the behaviour of some subset of \(Y^n\). To get their results, the authors introduce and study a particular vector basis of \(\mathbb{R}^n\).
As the main application of their results, the authors show that the pairs \((\ell_\infty^n,L_1(\mu))\) have the Bishop-Phelps-Bollobás property for every positive measure \(\mu\) and every positive integer~\(n\).
Reviewer: Miguel Martín (Granada)Toeplitz and asymptotic Toeplitz operators on \(H^2(\mathbb{D}^n)\).https://www.zbmath.org/1456.470082021-04-16T16:22:00+00:00"Maji, Amit"https://www.zbmath.org/authors/?q=ai:maji.amit"Sarkar, Jaydeb"https://www.zbmath.org/authors/?q=ai:sarkar.jaydeb"Sarkar, Srijan"https://www.zbmath.org/authors/?q=ai:sarkar.srijanSummary: We initiate a study of Toeplitz operators and asymptotic Toeplitz operators on the Hardy space \(H^2(\mathbb{D}^n)\) (over the unit polydisc \(\mathbb{D}^n\) in \(\mathbb{C}^n\)). Our main results on Toeplitz and asymptotic Toeplitz operators can be stated as follows: Let \(T_{z_i}\) denote the multiplication operator on \(H^2(\mathbb{D}^n)\) by the \(i\)-th coordinate function \(z_i\), \(i = 1, \dots, n\), and let \(T\) be a bounded linear operator on \(H^2(\mathbb{D}^n)\). Then the following hold: \begin{itemize}\item[(i)] \(T\) is a Toeplitz operator (that is, \(T = P_{H^2(\mathbb{D}^n)} M_\varphi |_{H^2(\mathbb{D}^n)}\), where \(M_\varphi\) is the Laurent operator on \(L^2(\mathbb{T}^n)\) for some \(\varphi \in L^\infty(\mathbb{T}^n)\)) if and only if \(T_{z_i}^\ast T T_{z_i} = T\) for all \(i = 1, \dots, n\).\item[(ii)] \(T\) is an asymptotic Toeplitz operator if and only if \(T = \text{Toeplitz} + \text{compact}\).
\end{itemize}
The case \(n = 1\) gives the well-known results of \textit{A. Brown} and \textit{P. R. Halmos} [J. Reine Angew. Math. 213, 89--102 (1963; Zbl 0116.32501)] and \textit{A. Feintuch} [Oper. Theory, Adv. Appl. 41, 241--254 (1989; Zbl 0676.47014)], respectively. We also present related results in the setting of vector-valued Hardy spaces over the unit disc.Hilbert matrix on spaces of Bergman-type.https://www.zbmath.org/1456.470112021-04-16T16:22:00+00:00"Jevtić, Miroljub"https://www.zbmath.org/authors/?q=ai:jevtic.miroljub"Karapetrović, Boban"https://www.zbmath.org/authors/?q=ai:karapetrovic.bobanSummary: It is well known (see [\textit{M. Jevtić} and \textit{B. Karapetrović}, ``Libera operator on mixed norm spaces \(H_\nu^{p, q, \alpha}\) when \(0 <p < 1\)'', Filomat 31, No. 14, 4641--4650 (2017; \url{doi:10.2298/FIL1714641J}); \textit{M. Pavlović}, ``Definition and properties of the Libera operator on mixed norm spaces'', The Scientific World Journal 2014, Article ID 590656, 15 p. (2014; \url{doi:10.1155/2014/590656})]) that the Libera operator \(\mathcal{L}\) is bounded on the Besov space \(H_\nu^{p, q, \alpha}\) if and only if \(0 < \kappa_{p, \alpha, \nu} : = \nu - \alpha - \frac{1}{p} + 1\). We prove unexpected results: the Hilbert matrix operator \(H\), as well as the modified Hilbert operator \(\tilde{H}\), is bounded on \(H_\nu^{p, q, \alpha}\) if and only if \(0 < \kappa_{p, \alpha, \nu} < 1\). In particular, \(H\), as well as \(\tilde{H}\), is bounded on the Bergman space \(A^{p, \alpha}\) if and only if \(1 < \alpha + 2 < p\) and is bounded on the Dirichlet space \(\mathcal{D}_\alpha^p = A_1^{p, \alpha}\) if and only if \(\max \{- 1, p - 2 \} < \alpha < 2 p - 2\). Our results are substantial improvement of [\textit{B. Łanucha} et al., Ann. Acad. Sci. Fenn., Math. 37, No. 1, 161--174 (2012; Zbl 1258.47047), Theorem 3.1] and of [\textit{P. Galanopoulos} et al., Ann. Acad. Sci. Fenn., Math. 39, No. 1, 231--258 (2014; Zbl 1297.47030), Theorem 5].Weighted boundedness of the fractional maximal operator and Riesz potential generated by Gegenbauer differential operator.https://www.zbmath.org/1456.420202021-04-16T16:22:00+00:00"Ibrahimov, Elman J."https://www.zbmath.org/authors/?q=ai:ibrahimov.elman-j"Guliyev, Vagif S."https://www.zbmath.org/authors/?q=ai:guliyev.vagif-sabir"Jafarova, Saadat A."https://www.zbmath.org/authors/?q=ai:jafarova.saadat-aSummary: In the paper we study the weighted \((L_{p,\omega,\lambda}, L_{q,\omega,\lambda}) \)-boundedness of the fractional maximal operator \(M_G^\alpha \) (\(G\) is a fractional maximal operator) and the Riesz potential (\(G\) is the Riesz potential) generated by the Gegenbauer differential operator
\[G_{\lambda}\equiv G=( x^{2}-1) ^{\frac{1}{2}-\lambda}\frac{d}{dx}( x^{2}-1) ^{\lambda +\frac{1}{2}}\frac{d}{dx}, \quad x\in (1,\infty ),\quad \lambda \in \left( 0,\frac{1}{2}\right).\]Weighted composition operators from Dirichlet-type spaces into Stević-type spaces.https://www.zbmath.org/1456.300932021-04-16T16:22:00+00:00"Zhu, Xiangling"https://www.zbmath.org/authors/?q=ai:zhu.xianglingSummary: The boundedness and compactness of weighted composition operators from Dirichlet-type spaces into Stević-type spaces are investigated in this paper. Some estimates for the essential norm of weighted composition operators are also given.Dimensional reduction and scattering formulation for even topological invariants.https://www.zbmath.org/1456.814942021-04-16T16:22:00+00:00"Schulz-Baldes, Hermann"https://www.zbmath.org/authors/?q=ai:schulz-baldes.hermann"Toniolo, Daniele"https://www.zbmath.org/authors/?q=ai:toniolo.danieleSummary: Strong invariants of even-dimensional topological insulators of independent Fermions are expressed in terms of an invertible operator on the Hilbert space over the boundary. It is given by the Cayley transform of the boundary restriction of the half-space resolvent. This dimensional reduction is routed in new representation for the \(K\)-theoretic exponential map. It allows to express the invariants via the reflection matrix at the Fermi energy, for the scattering set-up of a wire coupled to the half-space insulator.On singular perturbations of quantum dynamical semigroups.https://www.zbmath.org/1456.470142021-04-16T16:22:00+00:00"Holevo, A. S."https://www.zbmath.org/authors/?q=ai:holevo.alexander-sSummary: We consider two examples of quantum dynamical semigroups obtained by singular perturbations of a standard generator which are special case of unbounded completely positive perturbations studied in detail in [\textit{A. S. Holevo}, Izv. Math. 59, No. 6, 1311--1325 (1995; Zbl 0877.47026); translation from Izv. Ross. Akad. Nauk, Ser. Mat. 59, No. 6, 207--222 (1995)]. In Section~2, we propose a generalization of an example in [\textit{A. S. Holevo}, Izv. Math. 59, No. 6, 1311--1325 (1995; Zbl 0877.47026); translation from Izv. Ross. Akad. Nauk, Ser. Mat. 59, No. 6, 207--222 (1995)] aimed to give a positive answer to a conjecture of Arveson [loc.\,cit.]. In Section~3, we consider in greater detail an improved and simplified construction of a nonstandard dynamical semigroup outlined in our short communication [\textit{A. S. Kholevo}, Russ. Math. Surv. 51, No. 6, 1206--1207 (1996; Zbl 1042.46513); translation from Usp. Mat. Nauk 51, No. 6, 225--226 (1996)].Volterra integral operators from \(\mathcal{D}^p_{p-2+s}\) into \(F(p\lambda,p\lambda+s\lambda-2,q)\).https://www.zbmath.org/1456.300952021-04-16T16:22:00+00:00"Shen, Conghui"https://www.zbmath.org/authors/?q=ai:shen.conghui"Lou, Zengjian"https://www.zbmath.org/authors/?q=ai:lou.zengjian"Li, Songxiao"https://www.zbmath.org/authors/?q=ai:li.songxiaoSummary: Let \(1<p<\infty\), \(0<q<\infty\), \(0<s\), \(\lambda\leqslant 1\) such that \(q+s\lambda>1\). We characterize the boundedness and compactness of inclusion mapping from Dirichlet type spaces \(\mathscr{D}^p_{p-2+s}\) into tent spaces \(T_{p\lambda,q}(\mu)\). As an application, the boundedness of the Volterra operator \(T_g\), its companion operator \(I_g\) and the multiplication operator \(M_g\) from \(\mathscr{D}^p_{p-2+s}\) to \(F(p\lambda,p\lambda+s\lambda-2,q)\) are given. Furthermore, we study the essential norm and compactness of \(T_g\) and \(I_g\).Spectral analysis for discontinuous non-self-adjoint singular Dirac operators with eigenparameter dependent boundary condition.https://www.zbmath.org/1456.470152021-04-16T16:22:00+00:00"Li, Kun"https://www.zbmath.org/authors/?q=ai:li.kun.2"Sun, Jiong"https://www.zbmath.org/authors/?q=ai:sun.jiong"Hao, Xiaoling"https://www.zbmath.org/authors/?q=ai:hao.xiaoling"Bao, Qinglan"https://www.zbmath.org/authors/?q=ai:bao.qinglanSummary: In this paper, a discontinuous non-self-adjoint (dissipative) Dirac operator with eigenparameter dependent boundary condition, and with two singular endpoints is studied. The interface conditions are imposed on the discontinuous point. Firstly, we pass the considered problem to a maximal dissipative operator \(L_h\) by using operator theoretic formulation. The self-adjoint dilation \(\mathcal{T}_h\) of \(L_h\) in the space \(\mathcal{H}\) is constructed, furthermore, the incoming and outgoing representations of \(\mathcal{T}_h\) and functional model are also constructed, hence in light of Lax-Phillips theory, we derive the scattering matrix. Using the equivalence between scattering matrix and characteristic function, a completeness theorem on the eigenvectors and associated vectors of this dissipative operator is proved.An operator-valued \(T(1)\) theorem for symmetric singular integrals in UMD spaces.https://www.zbmath.org/1456.420172021-04-16T16:22:00+00:00"Hytönen, Tuomas"https://www.zbmath.org/authors/?q=ai:hytonen.tuomas-pSummary: The natural BMO (bounded mean oscillation) conditions suggested by scalar-valued results are known to be insufficient for the boundedness of operator-valued paraproducts. Accordingly, the boundedness of operator-valued singular integrals has only been available under versions of the classical ``\(T(1) \in \text{BMO}\)'' assumptions that are not easily checkable. Recently, \textit{G. Hong} et al. [J. Funct. Anal. 278, No. 7, Article ID 108420, 27 p. (2020; Zbl 07155094)] observed that the situation improves remarkably for singular integrals with a symmetry assumption, so that a classical \(T(1)\) criterion still guarantees their \(L^2\)-boundedness on Hilbert space -valued functions. Here, these results are extended to general UMD (unconditional martingale differences) spaces with the same natural BMO condition for symmetrised paraproducts, and requiring in addition only the usual replacement of uniform bounds by \(R\)-bounds in the case of general singular integrals. In particular, under these assumptions, we obtain boundedness results on non-commutative \(L^p\) spaces for all \(1 < p < \infty\), without the need to replace the domain or the target by a related non-commutative Hardy space as in the results of G. Hong et al. [loc. cit.] for \(p \neq 2\).\(H^p\)-boundedness of Hankel Hausdorff operator involving Hankel transformation.https://www.zbmath.org/1456.420242021-04-16T16:22:00+00:00"Upadhyay, S. K."https://www.zbmath.org/authors/?q=ai:upadhyay.santosh-kumar"Pandey, Ravi Shankar"https://www.zbmath.org/authors/?q=ai:pandey.ravi-shankar"Mohapatra, R. N."https://www.zbmath.org/authors/?q=ai:mohapatra.ram-narayan|mohapatra.rabindra-nSummary: In this paper the \(H^p(0,\infty)\)-boundedness of Hankel Hausdorff operator involving Hankel transformation is investigated by the method used by \textit{Y. Kanjin} [Stud. Math. 148, No. 1, 37--45 (2001; Zbl 1001.47018)]. Some properties related to Hankel Hausdorff operator on \(H^p(0,\infty)\) space are discussed.On extrapolation properties of Schatten-von Neumann classes.https://www.zbmath.org/1456.470052021-04-16T16:22:00+00:00"Lykov, K. V."https://www.zbmath.org/authors/?q=ai:lykov.konstantin-vSummary: For a certain special class of symmetric sequence spaces, we give an explicit relation between the interpolation and extrapolation representations. This relation is carried over to symmetrically normed ideals of compact operators.Invariant subspaces for commuting operators on a real Banach space.https://www.zbmath.org/1456.470032021-04-16T16:22:00+00:00"Lomonosov, V. I."https://www.zbmath.org/authors/?q=ai:lomonosov.victor-i"Shul'man, Viktor S."https://www.zbmath.org/authors/?q=ai:shulman.victor-sSummary: It is proved that the commutative algebra \(\mathcal A\) of operators on a reflexive real Banach space has an invariant subspace if each operator \(T\in\mathcal A\) satisfies the condition
\[
\| 1 - \varepsilon T^2\|_e \leqslant 1 + o(\varepsilon)\text{ as }\varepsilon \searrow 0,
\]
where \(\| \cdot\|_e\) denotes the essential norm. This implies the existence of an invariant subspace for any commutative family of essentially self-adjoint operators on a real Hilbert space.\(p\)-regularity and weights for operators between \(L^p\)-spaces.https://www.zbmath.org/1456.460262021-04-16T16:22:00+00:00"Sánchez Pérez, Enrique A."https://www.zbmath.org/authors/?q=ai:sanchez-perez.enrique-alfonso"Tradacete, Pedro"https://www.zbmath.org/authors/?q=ai:tradacete-perez.pedroSummary: We explore the connection between \(p\)-regular operators on Banach function spaces and weighted \(p\)-estimates. In particular, our results focus on the following problem. Given finite measure spaces \(\mu\) and \(\nu\), let \(T\) be an operator defined from a Banach function space \(X(\nu)\) and taking values on \(L^p (v d \mu)\) for \(v\) in certain family of weights \(V\subset L^1(\mu)_+\) we analyze the existence of a bounded family of weights \(W\subset L^1(\nu)_+\) such that for every \(v\in V\) there is \(w \in W\) in such a way that \(T:L^p(w d \nu) \to L^p(v d \mu)\) is continuous uniformly on \(V\). A condition for the existence of such a family is given in terms of \(p\)-regularity of the integration map associated to a certain vector measure induced by the operator \(T\).The regularity of inverses to Sobolev mappings and the theory of \(\mathcal{Q}_{q,p} \)-homeomorphisms.https://www.zbmath.org/1456.300462021-04-16T16:22:00+00:00"Vodopyanov, S. K."https://www.zbmath.org/authors/?q=ai:vodopyanov.serguei-kSummary: We prove that each homeomorphism \(\varphi:D\to D^{\prime}\) of Euclidean domains in \(\mathbb{R}^n\), \(n\geq 2 \), belonging to the Sobolev class \(W^1_{p,\operatorname{loc}}(D) \), where \(p\in[1,\infty) \), and having finite distortion induces a bounded composition operator from the weighted Sobolev space \(L^1_p(D^{\prime};\omega)\) into \(L^1_p(D)\) for some weight function \(\omega:D^{\prime}\to(0,\infty) \). This implies that in the cases \(p>n-1\) and \(n\geq 3\) as well as \(p\geq 1\) and \(n\geq 2\) the inverse \(\varphi^{-1}:D^{\prime}\to D\) belongs to the Sobolev class \(W^1_{1,\operatorname{loc}}(D^{\prime}) \), has finite distortion, and is differentiable \({\mathcal{H}}^n \)-almost everywhere in \(D^{\prime} \). We apply this result to \(\mathcal{Q}_{q,p} \)-homeomorphisms; the method of proof also works for homeomorphisms of Carnot groups. Moreover, we prove that the class of \(\mathcal{Q}_{q,p} \)-homeomorphisms is completely determined by the controlled variation of the capacity of cubical condensers whose shells are concentric cubes.On compact operators on some sequence spaces related to matrix \(B(r,s,t)\).https://www.zbmath.org/1456.470102021-04-16T16:22:00+00:00"Demiriz, Serkan"https://www.zbmath.org/authors/?q=ai:demiriz.serkan"Kara, Emrah Evren"https://www.zbmath.org/authors/?q=ai:kara.emrah-evrenSummary: In the present paper, we establish some identities or estimates for the operator norms and the Hausdorff measures of noncompactness of certain matrix operators on the spaces \(c_0(B)\), \(\ell_\infty(B)\) and \(\ell_{p}(B)\) which have recently been introduced in [\textit{A. Sönmez}, Comput. Math. Appl. 62, No. 2, 641--650 (2011; Zbl 1228.40006)]. Further, by using the Hausdorff measure of noncompactness, we characterize some classes of compact operators on these spaces.Hyperreflexivity constants of the bounded \(n\)-cocycle spaces of group algebras and \(C^*\)-algebras.https://www.zbmath.org/1456.460452021-04-16T16:22:00+00:00"Samei, Ebrahim"https://www.zbmath.org/authors/?q=ai:samei.ebrahim"Farsani, Jafar Soltani"https://www.zbmath.org/authors/?q=ai:farsani.jafar-soltaniSummary: We introduce the concept of strong property \((\mathbb{B})\) with a constant for Banach algebras and, by applying a certain analysis on the Fourier algebra of the unit circle, we show that all \(C^*\)-algebras and group algebras have the strong property \((\mathbb{B})\) with a constant given by \(288 \pi (1 + \sqrt{2})\). We then use this result to find a concrete upper bound for the hyperreflexivity constant of \(\mathcal{Z}^n (A, X)\), the space of bounded \(n\)-cocycles from \(A\) into \(X\), where \(A\) is a \(C^*\)-algebra or the group algebra of a group with an open subgroup of polynomial growth and \(X\) is a Banach \(A\)-bimodule for which \(\mathcal{H}^{n+1} (A, X)\) is a Banach space. As another application, we show that for a locally compact amenable group \(G\) and \(1 < p < \infty\), the space \(CV_P(G)\) of convolution operators on \(L^p(G)\) is hyperreflexive with a constant given by \(384 \pi^2 (1 + \sqrt{2})\). This is the generalization of a well-known result of \textit{E. Christensen} [Math. Scand. 50, 111--122 (1982; Zbl 0503.47032)] for \(p = 2\).Discrete Cesaro operator between weighted Banach spaces on homogenous trees.https://www.zbmath.org/1456.460212021-04-16T16:22:00+00:00"Sharma, Ajay K."https://www.zbmath.org/authors/?q=ai:sharma.ajay-kumar|sharma.ayay-k"Kumar, Vivek"https://www.zbmath.org/authors/?q=ai:kumar.vivekThis paper studies the discrete Cesàro operator on weighted Banach spaces on homogenous
trees. Continuity and compactness are characterized. The discrete Cesàro operator acting on the Banach space of bounded functions on homogenous trees is continuous, but not compact. This operator acting on weighted Banach spaces of bounded functions on homogenous
trees may not be continuous for some weight functions. Moreover, examples of weight functions are presented such that the operator acting on the corresponding weighted Banach spaces of bounded functions on homogenous trees is compact.
Reviewer: José Bonet (Valencia)Geometry of \(\ell_p^n\)-balls: classical results and recent developments.https://www.zbmath.org/1456.460132021-04-16T16:22:00+00:00"Prochno, Joscha"https://www.zbmath.org/authors/?q=ai:prochno.joscha"Thäle, Christoph"https://www.zbmath.org/authors/?q=ai:thale.christoph"Turchi, Nicola"https://www.zbmath.org/authors/?q=ai:turchi.nicolaSummary: In this article we first review some by-now classical results about the geometry of \(\ell_p\)-balls \(\mathbb{B}_p^n\) in \(\mathbb{R}^n\) and provide modern probabilistic arguments for them. We also present some more recent developments including a central limit theorem and a large deviations principle for the \(q\)-norm of a random point in \(\mathbb{B}_p^n\). We discuss their relation to the classical results and give hints to various extensions that are available in the existing literature.
For the entire collection see [Zbl 1431.60003].Spectral properties of Toeplitz operators on the unit ball and on the unit sphere.https://www.zbmath.org/1456.470062021-04-16T16:22:00+00:00"Akkar, Zineb"https://www.zbmath.org/authors/?q=ai:akkar.zineb"Albrecht, Ernst"https://www.zbmath.org/authors/?q=ai:albrecht.ernstSummary: In this article, we consider Toeplitz operators on the Hardy and weighted Bergman Hilbert spaces of the unit sphere, respectively on the unit ball in $\mathbb{C}^N$. Various aspects of the interplay between local and global properties of the symbols and local and global spectral properties of the corresponding Toeplitz operators are investigated. A local version of the spectral inclusion theorem of \textit{A. M. Davie} and \textit{N. P. Jewell} [J. Funct. Anal. 26, 356--368 (1977; Zbl 0374.47011)] is proved. Using some recent results of \textit{R. Quiroga-Barranco} and \textit{N. Vasilevski} [Integral Equations Oper. Theory 59, No. 3, 379--419 (2007; Zbl 1144.47024); ibid. 60, No. 1, 89--132 (2008; Zbl 1144.47025)], we describe some commutative $C^*$-subalgebras of the Toeplitz algebra for $N \geq2$. The method of \textit{G. McDonald} [Ill. J. Math. 23, 286--293 (1979; Zbl 0438.47031)] to compute the essential spectrum of Toeplitz operators with certain piecewise continuous symbols is extended to a larger class of symbols including examples where the surface measure of set of discontinuity points has strictly positive measure.
For the entire collection see [Zbl 1300.47008].Derivations and automorphisms of nilpotent evolution algebras with maximal nilindex.https://www.zbmath.org/1456.170192021-04-16T16:22:00+00:00"Mukhamedov, Farrukh"https://www.zbmath.org/authors/?q=ai:mukhamedov.farruh-m"Khakimov, Otabek"https://www.zbmath.org/authors/?q=ai:khakimov.otabek-n"Omirov, Bakhrom"https://www.zbmath.org/authors/?q=ai:omirov.bakhrom-a"Qaralleh, Izzat"https://www.zbmath.org/authors/?q=ai:qaralleh.izzatAn algebra \((E,+,\cdot)\) over a field \(K\) is called an \textit{evolution algebra} provided it has a basis \(\{e_i\}\) such that \(e_i\cdot e_j=\) for everi \(i\neq j\) and \(e_i\cdot e_i=\sum_k a_{i,k}e_k\). Such a basis, is called \textit{natural basis}. This paper deals with nilpotent evolution algebras \(E\) such that \(\dim E^2= \dim E -1\).
The authors describe the Lie algebra of derivations of \(E\) when \(E\) has maximal index of nilpotency. Furthermore, they describe local and 2-local derivations for such algebras showing, for example, that every 2-local derivation of \(E\) is a derivation.
Finally, the authors determine the automorphisms and local automorphims for this type of algebras.
Reviewer: Antonio M. Oller Marcén (Zaragoza)Operators on weighted Lorentz-Karamata-Bochner spaces.https://www.zbmath.org/1456.470092021-04-16T16:22:00+00:00"Datt, Gopal"https://www.zbmath.org/authors/?q=ai:datt.gopal.1"Chugh, Renu"https://www.zbmath.org/authors/?q=ai:chugh.renu"Jakhar, Jagjeet"https://www.zbmath.org/authors/?q=ai:jakhar.jagjeetSummary: In this paper, conditions are derived for the multiplication, composition and weighted composition operators on weighted Lorentz-Karamata-Bochner (WLKB) spaces \(L_{p,q,w;b}\), \(1 < p\leq \infty\), \(1\leq q\leq\infty\), to be bounded.Spectral analysis of the Laplacian acting on discrete cusps and funnels.https://www.zbmath.org/1456.811752021-04-16T16:22:00+00:00"Athmouni, Nassim"https://www.zbmath.org/authors/?q=ai:athmouni.nassim"Ennaceur, Marwa"https://www.zbmath.org/authors/?q=ai:ennaceur.marwa"Golénia, Sylvain"https://www.zbmath.org/authors/?q=ai:golenia.sylvainSummary: We study perturbations of the discrete Laplacian associated to discrete analogs of cusps and funnels. We perturb the metric and the potential in a long-range way. We establish a propagation estimate and a Limiting Absorption Principle away from the possible embedded eigenvalues. The approach is based on a positive commutator technique.Biduality in weighted spaces of analytic functions.https://www.zbmath.org/1456.460142021-04-16T16:22:00+00:00"Boyd, Christopher"https://www.zbmath.org/authors/?q=ai:boyd.christopher"Rueda, Pilar"https://www.zbmath.org/authors/?q=ai:rueda.pilarSummary: We study new conditions for non necessarily radial weights implying that the weighted Banach space \(\mathcal{H}_v(U)\) of analytic functions \(f\) such that \(vf\) is bounded on \(U\), is canonically isometrically isomorphic to the bidual of \(\mathcal{H}_{v_o}(U)\), its closed subspace formed by those functions \(f\) such that \(vf\) converges to \(0\) on the boundary of \(U\). We provide several examples of weights that satisfy these conditions. As an application, we show that whenever \(\mathcal{H}_v(U)=\mathcal{H}_{v_o}(U)''\) the norm-attaining functions are dense in \(\mathcal{H}_v(U)\).
For the entire collection see [Zbl 1444.15003].A product expansion for Toeplitz operators on the Fock space.https://www.zbmath.org/1456.470072021-04-16T16:22:00+00:00"Hagger, Raffael"https://www.zbmath.org/authors/?q=ai:hagger.raffaelSummary: We study the asymptotic expansion of the product of two Toeplitz operators on the Fock space. In comparison to earlier results, we require significantly fewer derivatives and get the expansion to arbitrary order. This, in particular, improves a result of Borthwick related to Toeplitz quantization [\textit{D. Borthwick}, Contemp. Math. 214, 23--37 (1998; Zbl 0903.58013)]. In addition, we derive an intertwining identity between the Berezin star product and the sharp product.Spin-boson type models analyzed using symmetries.https://www.zbmath.org/1456.811782021-04-16T16:22:00+00:00"Dam, Thomas Norman"https://www.zbmath.org/authors/?q=ai:dam.thomas-norman"Møller, Jacob Schach"https://www.zbmath.org/authors/?q=ai:schach-moller.jacobThis paper is devoted to the investigation of a family of models for a qubit interacting with a bosonic field. The authors consider state space \(\mathbb{C}^2 \otimes\mathcal{F}_b(\mathcal{H})\), where Hilbert space \(\mathcal{H}\) is the state space of a single boson and \(\mathcal{F}_b(\mathcal{H})\) is the corresponding bosonic Fock space; the state space of the qubit is \(\mathbb{C}^2\). In the paper under review the following Hamiltonian is investigated
\[H_\eta(\alpha,f,\omega)= \eta\sigma_z\otimes 1 + 1\otimes d\Gamma(\omega)+\sum\limits_{i=1}^{2n} \alpha_i \left(\sigma_x\otimes \phi (f_i)\right)^i .\]
This operator is parameterized by \(\alpha\in\mathbb{C}^{2n}, f\in\mathcal{H}^{2n}, \eta\in\mathbb{C}\), \(\sigma_x, \sigma_y, \sigma_z\) denote the Pauli matrices, \(\omega\) is self-adjoint on \(\mathcal{H}\) and \(d\Gamma(\omega)\) is the second quantization of \(\omega\).
It is assumed that this Hamiltonian has a special symmetry, called \textit{spin-parity symmetry}. The spin-parity symmetry allows to find the domain of self-adjointness and decompose the Hamiltonian into two fiber operators each defined on Fock space. The authors prove the Hunziker-van Winter-Zhislin (HVZ) theorem for the fiber operators. The HVZ theorem for the fiber operators also gives an HVZ theorem for the full Hamiltonian. It is proved that if ground states exist for the full Hamiltonian, then the bottom of the spectrum is a nondegenerate
eigenvalue. Using this result, the authors single out a particular fiber, which has a ground state if and only if the full Hamiltonian has a ground state. Ground states for the other fiber operator must therefore correspond to excited states. A criterion for the existence of an excited state is also obtained.
Reviewer: Michael Perelmuter (Kyjiw)Characterizations of \({*}\) and \({*}\)-left derivable mappings on some algebras.https://www.zbmath.org/1456.460562021-04-16T16:22:00+00:00"An, Guangyu"https://www.zbmath.org/authors/?q=ai:an.guangyu"He, Jun"https://www.zbmath.org/authors/?q=ai:he.jun"Li, Jiankui"https://www.zbmath.org/authors/?q=ai:li.jiankuiSummary: A linear mapping \(\delta\) from a \({*}\)-algebra \(\mathcal{A}\) into a \({*}$-$\mathcal{A} \)-bimodule \(\mathcal{M}\) is a \({*} \)-derivable mapping at \(G\in \mathcal{A}\) if \(A\delta (B)^*+\delta (A)B=\delta (G)\) for each \(A, B\) in \(\mathcal{A}\) with \(AB^*=G\). We prove that every (continuous) \({*} \)-derivable mapping at \(G\) from a (unital \(C^*\)-algebra) factor von Neumann algebra into its Banach \({*}\)-bimodule is a \({*}\)-derivation if and only if \(G\) is a left separating point. A linear mapping \(\delta\) from a \({*}\)-algebra \(\mathcal{A}\) into a \({*} \)-left \(\mathcal{A} \)-module \(\mathcal{M}\) is a \({*} \)-left derivable mapping at \(G\in \mathcal{A}\) if \(A\delta (B)^*+B\delta (A)=\delta (G)\) for each \(A, B\) in \(\mathcal{A}\) with \(AB^*=G\). We prove that every continuous \({*} \)-left derivable mapping at a left separating point from a unital \(C^*\)-algebra or von Neumann algebra into its Banach \({*}\)-left \(\mathcal{A} \)-module is identical with zero under certain conditions.On self-adjoint extensions and symmetries in quantum mechanics.https://www.zbmath.org/1456.811792021-04-16T16:22:00+00:00"Ibort, Alberto"https://www.zbmath.org/authors/?q=ai:ibort.alberto"Lledó, Fernando"https://www.zbmath.org/authors/?q=ai:lledo.fernando"Pérez-Pardo, Juan Manuel"https://www.zbmath.org/authors/?q=ai:perez-pardo.juan-manuelSummary: Given a unitary representation of a Lie group \(G\) on a Hilbert space \(\mathcal H\), we develop the theory of \(G\)-invariant self-adjoint extensions of symmetric operators using both von Neumann's theorem and the theory of quadratic forms. We also analyze the relation between the reduction theory of the unitary representation and the reduction of the \(G\)-invariant unbounded operator. We also prove a \(G\)-invariant version of the representation theorem for closed and semi-bounded quadratic forms. The previous results are applied to the study of \(G\)-invariant self-adjoint extensions of the Laplace-Beltrami operator on a smooth Riemannian manifold with boundary on which the group \(G\) acts. These extensions are labeled by admissible unitaries \(U\) acting on the \(L^2\)-space at the boundary and having spectral gap at \(-1\). It is shown that if the unitary representation \(V\) of the symmetry group \(G\) is traceable, then the self-adjoint extension of the Laplace-Beltrami operator determined by \(U\) is \(G\)-invariant if \(U\) and \(V\) commute at the boundary. Various significant examples are discussed at the end.Study of the algebra of smooth integro-differential operators with applications.https://www.zbmath.org/1456.160212021-04-16T16:22:00+00:00"Haghany, A."https://www.zbmath.org/authors/?q=ai:haghany.ahmad"Kassaian, Adel"https://www.zbmath.org/authors/?q=ai:kassaian.adelComplete characterization of bounded composition operators on the general weighted Hilbert spaces of entire Dirichlet series.https://www.zbmath.org/1456.300032021-04-16T16:22:00+00:00"Doan, Minh Luan"https://www.zbmath.org/authors/?q=ai:doan.minh-luan"Lê, Hai Khôi"https://www.zbmath.org/authors/?q=ai:le.hai-khoiSummary: We establish necessary and sufficient conditions for boundedness of composition operators on the most general class of Hilbert spaces of entire Dirichlet series with real frequencies. Depending on whether or not the space being considered contains any nonzero constant function, different criteria for boundedness are developed. Thus, we complete the characterization of bounded composition operators on all known Hilbert spaces of entire Dirichlet series of one variable.Sklyanin-like algebras for \((q\)-)linear grids and \((q\)-)para-Krawtchouk polynomials.https://www.zbmath.org/1456.812652021-04-16T16:22:00+00:00"Bergeron, Geoffroy"https://www.zbmath.org/authors/?q=ai:bergeron.geoffroy"Gaboriaud, Julien"https://www.zbmath.org/authors/?q=ai:gaboriaud.julien"Vinet, Luc"https://www.zbmath.org/authors/?q=ai:vinet.luc"Zhedanov, Alexei"https://www.zbmath.org/authors/?q=ai:zhedanov.alexei-sSummary: S-Heun operators on linear and \(q\)-linear grids are introduced. These operators are special cases of Heun operators and are related to Sklyanin-like algebras. The continuous Hahn and big \(q\)-Jacobi polynomials are functions on which these S-Heun operators have natural actions. We show that the S-Heun operators encompass both the bispectral operators and Kalnins and Miller's structure operators. These four structure operators realize special limit cases of the trigonometric degeneration of the original Sklyanin algebra. Finite-dimensional representations of these algebras are obtained from a truncation condition. The corresponding representation bases are finite families of polynomials: the para-Krawtchouk and \(q\)-para-Krawtchouk ones. A natural algebraic interpretation of these polynomials that had been missing is thus obtained. We also recover the Heun operators attached to the corresponding bispectral problems as quadratic combinations of the S-Heun operators.
{\copyright 2021 American Institute of Physics}Generic zero-Hausdorff and one-packing spectral measures.https://www.zbmath.org/1456.811772021-04-16T16:22:00+00:00"Carvalho, Silas L."https://www.zbmath.org/authors/?q=ai:carvalho.silas-l"de Oliveira, César R."https://www.zbmath.org/authors/?q=ai:de-oliveira.cesar-rSummary: For some metric spaces of self-adjoint operators, it is shown that the set of operators whose spectral measures have simultaneous zero upper-Hausdorff and one lower-packing dimension contains a dense \(G_\delta\) subset. Applications include sets of limit-periodic operators.
{\copyright 2021 American Institute of Physics}Asymptotics of eigenvalues and eigenvectors of Toeplitz matrices.https://www.zbmath.org/1456.150052021-04-16T16:22:00+00:00"Dai, Hui"https://www.zbmath.org/authors/?q=ai:dai.hui"Geary, Zachary"https://www.zbmath.org/authors/?q=ai:geary.zachary"Kadanoff, Leo P."https://www.zbmath.org/authors/?q=ai:kadanoff.leo-pOn the asymptotic dynamics of 2-D magnetic quantum systems.https://www.zbmath.org/1456.811762021-04-16T16:22:00+00:00"Cárdenas, Esteban"https://www.zbmath.org/authors/?q=ai:cardenas.esteban"Hundertmark, Dirk"https://www.zbmath.org/authors/?q=ai:hundertmark.dirk"Stockmeyer, Edgardo"https://www.zbmath.org/authors/?q=ai:stockmeyer.edgardo"Vugalter, Semjon"https://www.zbmath.org/authors/?q=ai:vugalter.semjon-aSummary: In this work, we provide results on the long-time localization in space (dynamical localization) of certain two-dimensional magnetic quantum systems. The underlying Hamiltonian may have the form \(H = H_0 + W\), where \(H_0\) is rotationally symmetric and has dense point spectrum and \(W\) is a perturbation that breaks the rotational symmetry. In the latter case, we also give estimates for the growth of the angular momentum operator in time.The joint numerical radius on \(C^*\)-algebras.https://www.zbmath.org/1456.460432021-04-16T16:22:00+00:00"Mabrouk, Mohamed"https://www.zbmath.org/authors/?q=ai:mabrouk.mohamedSummary: Let \(\mathfrak{A}\) be unital \(C^*\)-algebra with unit \(e\) and positive cone \(\mathfrak{A}^+\) such that every irreducible representation is infinite dimensional. For every \(\mathbf{a} =(a_1,\dots,a_n)\in\mathfrak{A}^n\), the joint numerical radius of \(\mathbf{a}\) is denoted by \(\mathbf{v}(\mathbf{a})\). It is shown that an element \(\mathbf{a}\in \mathfrak{A}^n\) satisfies \(\sum_{j=1}^n|f(a_j)|^2=1\) for every pure state \(f\) of \(\mathfrak{A}\) if and only if each \(a_j\) is in the center of \(\mathfrak{A}\) and \( \sum_{j=1}^na_j a_j^*=e\). Furthermore, we characterize elements \(\mathbf{a}_1, \dots, \mathbf{a}_n \in \mathfrak{A}^n\) such that for any \(\mathbf{x}\in (\mathfrak{A}^+)^n\) there exists \(\alpha =(\alpha_1, \dots, \alpha_n)\in \mathbb{R}^n\) such that \( \sum_{j=1}^{j=n}\alpha_j^2=1\) and \(\mathbf{v}(\sum_{j=1}^{j=n}\alpha_j\mathbf{a}_j+ \mathbf{x}) = 1+\mathbf{v}(\mathbf{x})\).
For the entire collection see [Zbl 1444.15003].Discrete self-adjointness and quantum dynamics. Travel times.https://www.zbmath.org/1456.811732021-04-16T16:22:00+00:00"Martínez-Pérez, Armando"https://www.zbmath.org/authors/?q=ai:martinez-perez.armando"Torres-Vega, Gabino"https://www.zbmath.org/authors/?q=ai:torres-vega.gabinoSummary: We use a discrete derivative to introduce a time operator for non-relativistic quantum systems with point spectrum. The symmetry requirement on the time operator leads to well-defined time values related to the dynamics of discrete quantum systems. As an illustration, we find travel times between hits with the walls for the quantum particle in a box model. These times suggest a classical analog of time eigenstates. We also briefly consider the Woods-Saxon potential and propose classical analogs for it.
{\copyright 2021 American Institute of Physics}