Characterizations of operator Birkhoff-James orthogonality

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