Ali Zamani. Source: Annals of Functional Analysis, Volume 10, Number 3, 433--445.Abstract:
In the following we generalize the concept of Birkhoff–James orthogonality of operators on a Hilbert space 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 $T\perp ^{B}_{A}S$ is defined if $T$ and $S$ are bounded with respect to the seminorm induced by a positive operator $A$ satisfying ${\|T+\gamma S\|}_{A}\geq {\|T\|}_{A}$ for all $\gamma \in \mathbb{C}$ . We extend a theorem due to Bhatia and Šemrl by proving that $T\perp ^{B}_{A}S$ if and only if there exists a sequence of $A$ -unit vectors $\{x_{n}\}$ in $\mathcal{H}$ such that $\lim _{n\rightarrow +\infty }{\|Tx_{n}\|}_{A}={\|T\|}_{A}$ and $\lim _{n\rightarrow +\infty }{\langle Tx_{n},Sx_{n}\rangle }_{A}=0$ . In addition, we give some $A$ -distance formulas. Particularly, we prove
\[\inf _{\gamma \in \mathbb{C}}{\Vert T+\gamma S\Vert }_{A}=\sup \{\vert {\langle Tx,y\rangle }_{A}\vert ;{\Vert x\Vert }_{A}={\Vert y\Vert }_{A}=1,{\langle Sx,y\rangle }_{A}=0\}.\] Some other related results are also discussed.