Birkhoff--James orthogonality of operators in semi-Hilbertian spaces and its applications

Abstract

In this paper, the concept of Birkhoff--James orthogonality of operators on a Hilbert space is generalized when a semi-inner product is considered. More precisely, for linear operators $T$ and $S$ on a complex Hilbert space $\mathcal{H}$, a new relation $T{\perp }_A^{B}S$ is defined if $T$ and $S$ are bounded with respect to the seminorm induced by a positive operator $A$ satisfying ${\Vert T+\gamma S\Vert }_A\ge {\Vert T\Vert }_A$ for all $\gamma \in \mathbb{C}$. We extend a theorem due to R. Bhatia and P. Šemrl, by proving that $T{\perp }_A^{B}S$ if and only if there exists a sequence of $A$-unit vectors $\{x_n\}$ in $\mathcal{H}$ such that $\underset{n\to +\infty }{\lim }{\Vert T{x}_n\Vert }_A={\Vert T\Vert }_A$ and $\underset{n\to +\infty }{\lim }{⟨T{x}_n,S{x}_n⟩}_A=0$. In addition, we give some $A$-distance formulas. Particularly, we prove \begin{align*} \displaystyle{\inf_{γ\in \mathbb{C}}}{\|T + γS\|}_{A} = \sup\Big\{|{\langle Tx, y\rangle}_A|; \, {\|x\|}_{A} = {\|y\|}_{A} = 1, \, {\langle Sx, y\rangle}_A = 0\Big\}. \end{align*} Some other related results are also discussed.

arXiv comment13 pages, to appear in Ann. Funct. Anal. (AFA)