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 $T\perp^B_A S$ 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 R. Bhatia and P. \v{ S}emrl, 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 $\displaystyle{ \lim_{ n\rightarrow +\infty}}{ \|Tx_n\|}_A = { \|T\|}_A$ and $\displaystyle{ \lim_{ n\rightarrow +\infty}}{ \langle Tx_n, Sx_n\rangle}_A = 0$. In addition, we give some $A$-distance formulas. Particularly, we prove \begin{ align*} \displaystyle{ \inf_{ \gamma \in \mathbb{ C}}}{ \|T + \gamma S\|}_{ 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.
Categories: math.FA