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Characterizations of Operator Birkhoff–James Orthogonality

Published online by Cambridge University Press:  20 November 2018

Mohammad Sal Moslehian  and
Ali Zamani
Mohammad Sal Moslehian
Affiliation:
Department of Pure Mathematics, Ferdowsi University of Mashhad, P.O. Box 1159, Mashhad 91775, Iran e-mail: moslehian@um.ac.ir
Ali Zamani
Affiliation:
Department of Mathematics, Farhangian University, Iran e-mail: zamani.ali85@yahoo.com
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Abstract

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In this paper, we obtain some characterizations of the (strong) Birkhoff–James orthogonality for elements of Hilbert ${{C}^{*}}$-modules and certain elements of $\mathbb{B}\left( H \right)$. Moreover, we obtain a kind of Pythagorean relation for bounded linear operators. In addition, for $T\in \mathbb{B}(H)$ we prove that if the norm attaining set ${{\mathbb{M}}_{T}}$ is a unit sphere of some finite dimensional subspace ${{H}_{0}}$ of $H$ and $||T|{{|}_{{{H}_{0}}\bot }}\,<\,\,||T||$, then for every $S\in \mathbb{B}(H)$, $T$ is the strong Birkhoff–James orthogonal to $S$ if and only if there exists a unit vector $\xi \in {{H}_{0}}$ such that $||T||\xi =\,|T|\xi $ and ${{S}^{*}}T\xi =0$. Finally, we introduce a new type of approximate orthogonality and investigate this notion in the setting of inner product ${{C}^{*}}$-modules.

Keywords

46L0546L0846B20Hilbert C*-moduleBirkhoÒ–James orthogonalitystrong BirkhoÒ–James orthogonalityapproximate orthogonality
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 60 , Issue 4 , 01 December 2017 , pp. 816 - 829
Copyright
Copyright © Canadian Mathematical Society 2017

References

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