Characterizations of Operator Birkhoff–James Orthogonality

Moslehian, Mohammad Sal ; Zamani, Ali

Canadian mathematical bulletin, Tome 60 (2017) no. 4, pp. 816-829

Résumé

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.

Détail

DOI : 10.4153/CMB-2017-004-5

Mots-clés : 46L05, 46L08, 46B20, Hilbert C*-module, BirkhoÒ–James orthogonality, strong BirkhoÒ–James orthogonality, approximate orthogonality