On symmetry of strong Birkhoff orthogonality in \(B(\mathcal{H},\mathcal{K})\) and \(K(\mathcal{H},\mathcal{K})\)
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.
scientific article; zbMATH DE number 7222420
scientific article