On symmetry of strong Birkhoff orthogonality in \(B(\mathcal{H},\mathcal{K})\) and \(K(\mathcal{H},\mathcal{K})\)
type: Item
instance of: scholarly article
title: On symmetry of strong Birkhoff orthogonality in \(B(\mathcal{H},\mathcal{K})\) and \(K(\mathcal{H},\mathcal{K})\)
zbMATH Open document ID: 1461.46011
author: Ryotaro Tanaka
author: Debmalya Sain
published in: Annals of Functional Analysis
publication date: 2020-07-17T00:00:00Z
review text: The authors provide a complete characterization of symmetry of strong Birkhoff orthogonality in \(B(\mathcal{H},\mathcal{K})\) and \(K(\mathcal{H},\mathcal{K})\) by considering \(B(\mathcal{H},\mathcal{K})\) (\(K(\mathcal{H},\mathcal{K})\)) as Hilbert \(C^*\)-module over \(B(\mathcal{H})$ $(K(\mathcal{H}))\) under the \(B(\mathcal{H})\) \((K(\mathcal{H}))\)-valued inner product \( \langle \;{,}\; \rangle \) given by \( \langle S, T \rangle = S^*T \) for each \( S, T \in B(\mathcal{H}, \mathcal{K})$ $(K(\mathcal{H},\mathcal{K}))\). The results obtained here are a nice continuation/generalization of the works by \textit{P. Ghosh} et al. [Linear Algebra Appl. 500, 43--51 (2016; Zbl 1350.46014)] and \textit{D. Sain} et al. [Oper. Matrices 11, No. 4, 1087--1095 (2017; Zbl 1465.47014)]. The authors prove the following: \(S \in B(\mathcal{H},\mathcal{K})\) is right symmetric if and only if (i) \(S\) is right invertible under the assumptions that \(\mathcal{H}\) is infinite-dimensional or \(\dim\mathcal{H}\geq\dim\mathcal{K}\) and (ii) \(S\) is a scalar multiple of an isometry under the assumption that \(\mathcal{H}\) is finite-dimensional and \(\dim \mathcal{H} < \dim \mathcal{K}\). Considering \(S \in K(\mathcal{H},\mathcal{K})\) they prove that \(S\) is right symmetric if and only if \(S\) has dense range. They also prove that \(S \in B(\mathcal{H},\mathcal{K})\) \( (K(\mathcal{H},\mathcal{K}))\) is left symmetric if and only if \(S\) is of rank one.
reviewed by: Kallol Paul
Mathematics Subject Classification ID: 46B20
Mathematics Subject Classification ID: 46L05
Mathematics Subject Classification ID: 46L08
Mathematics Subject Classification ID: 47A05
zbMATH DE Number: 7222420
zbMATH Keywords: (strong) Birkhoff orthogonality
zbMATH Keywords: symmetric point
zbMATH Keywords: bounded linear operator
zbMATH Keywords: compact operator
MaRDI profile type: Publication
full work available at URL: s43034-019-00048-7
OpenAlex ID: W3025156967
cites work: On Birkhoff orthogonality and isosceles orthogonality in normed linear spaces
cites work: Characterizations of inner product spaces
cites work: The Birkhoff-James orthogonality in Hilbert \(C^{\ast }\)-modules
cites work: A strong version of the Birkhoff-James orthogonality in Hilbert \(C^\ast\)-modules
cites work: Operators preserving the strong Birkhoff-James orthogonality on \(\mathbf{B}(H)\)
cites work: On symmetry of the (strong) Birkhoff-James orthogonality in Hilbert \(C^*\)-modules
cites work: Orthogonality of matrices and some distance problems
cites work: Orthogonality in linear metric spaces
cites work: Operator algebras. Theory of \(C^*\)-algebras and von Neumann algebras
cites work: Approximate symmetry of Birkhoff orthogonality
cites work: Symmetric properties of orthogonality of linear operators on (R^n, ||.||_1)
cites work: Orthogonality of bounded linear operators
cites work: On symmetry of Birkhoff-James orthogonality of linear operators
cites work: Orthogonality and Linear Functionals in Normed Linear Spaces
cites work: Inner products in normed linear spaces
cites work: Symmetric points for (strong) Birkhoff orthogonality in von Neumann algebras with applications to preserver problems
cites work: LEFT SYMMETRIC POINTS FOR BIRKHOFF ORTHOGONALITY IN THE PREDUALS OF VON NEUMANN ALGEBRAS
cites work: On symmetry of Birkhoff orthogonality in the positive cones of \(C^{\ast}\)-algebras with applications
cites work: Symmetry of Birkhoff-James orthogonality of operators defined between infinite dimensional Banach spaces
cites work: Q5403624
cites work: Birkhoff-James orthogonality and smoothness of bounded linear operators
cites work: Orthogonality of bounded linear operators on complex Banach spaces
cites work: Birkhoff-James orthogonality of linear operators on finite dimensional Banach spaces
cites work: On symmetry of Birkhoff-James orthogonality of linear operators on finite-dimensional real Banach spaces
cites work: Operator norm attainment and Birkhoff-James orthogonality
cites work: A complete characterization of Birkhoff-James orthogonality in infinite dimensional normed space
cites work: On operators preserving James' orthogonality
cites work: A remark on orthogonality and symmetry of operators in \(\mathcal{B}(\mathcal{H})\)
DOI: 10.1007/S43034-019-00048-7
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description: scientific article; zbMATH DE number 7222420