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

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 $T{\perp }_A^{B}S$ 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 Bhatia and 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 ${lim\hspace{0.17em}}_{n\to +\infty }\Vert T{x}_n{\Vert }_A=\Vert T{\Vert }_A$ and ${lim\hspace{0.17em}}_{n\to +\infty }\langle T{x}_n,S{x}_n{\rangle }_A=0$. In addition, we give some $A$-distance formulas. Particularly, we prove

$${inf\hspace{0.17em}}_{\gamma \in \mathbb{C}}\Vert T+\gamma S{\Vert }_A=sup\hspace{0.17em}\left\{\right|\langle Tx,y{\rangle }_A|;\Vert x{\Vert }_A=\Vert y{\Vert }_A=1,\langle Sx,y{\rangle }_A=0\}.$$ Some other related results are also discussed.