Characterizations of operator Birkhoff-James orthogonality

Abstract

In this paper, we obtain some characterizations of the (strong) Birkhoff--James orthogonality for elements of Hilbert $C^{\ast }$-modules and certain elements of $\mathbb{B}(\mathcal{H})$. Moreover, we obtain a kind of Pythagorean relation for bounded linear operators. In addition, for $T\in \mathbb{B}(\mathcal{H})$ we prove that if the norm attaining set ${\mathbb{M}}_T$ is a unit sphere of some finite dimensional subspace ${\mathcal{H}}_0$ of $\mathcal{H}$ and $\Vert T{\Vert }_{{{\mathcal{H}}_0}^{\perp }}<\Vert T\Vert$, then for every $S\in \mathbb{B}(\mathcal{H})$, $T$ is the strong Birkhoff--James orthogonal to $S$ if and only if there exists a unit vector $\xi \in {\mathcal{H}}_0$ such that $\Vert T\Vert \xi =|T|\xi$ and $S^{\ast }T\xi =0$. Finally, we introduce a new type of approximate orthogonality and investigate this notion in the setting of inner product $C^{\ast }$-modules.

14 pages, to appear in Canad. Math. Bull