Canadian Mathematical Bulletin, ( ISI ), Volume (60), No (4), Year (2017-3) , Pages (816-829)

Title : ( Characterizations of operator Birkhoff–-James orthogonality )

Authors: Mohammad Sal Moslehian , Ali Zamani ,

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Abstract

In this paper, we obtain some characterizations of the (strong) Birkhoff--James orthogonality for elements of Hilbert $C^*$-modules and certain elements of $\B(\mathscr{H})$. Moreover, we obtain a kind of Pythagorean relation for bounded linear operators. In addition, for $T\in \B(\mathscr{H})$ we prove that if the norm attaining set $\mathbb{M}_T$ is a unit sphere of some finite dimensional subspace $\mathscr{H}_0$ of $\mathscr{H}$ and $\|T\|_{{{\mathscr{H}}_0}^\perp} < \|T\|$, then for every $S\in\B(\mathscr{H})$, $T$ is the strong Birkhoff--James orthogonal to $S$ if and only if there exists a unit vector $\xi\in {\mathscr{H}}_0$ 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

, Hilbert $C^*$, module; Birkhoff, , James orthogonality; strong Birkhoff, , James orthogonality; approximate orthogonality
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@article{paperid:1061847,
author = {Sal Moslehian, Mohammad and Ali Zamani},
title = {Characterizations of operator Birkhoff–-James orthogonality},
journal = {Canadian Mathematical Bulletin},
year = {2017},
volume = {60},
number = {4},
month = {March},
issn = {0008-4395},
pages = {816--829},
numpages = {13},
keywords = {Hilbert $C^*$-module; Birkhoff--James orthogonality; strong Birkhoff--James orthogonality; approximate orthogonality},
}

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%0 Journal Article
%T Characterizations of operator Birkhoff–-James orthogonality
%A Sal Moslehian, Mohammad
%A Ali Zamani
%J Canadian Mathematical Bulletin
%@ 0008-4395
%D 2017

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