Characterizations of operator Birkhoff-James orthogonality

In this paper, we obtain some characterizations of the (strong) Birkhoff–James orthogonality for elements of Hilbert $C^{\ast }$-modules and certain elements of $\mathbb{B}(\mathcal{H})$. Moreover, we obtain a kind of Pythagorean relation for bounded linear operators. In addition, for $T\in $\mathbb{B}$(\mathscr{H})$weprovethatifthenormattainingset$$\mathbb{M}$_T$isaunitsphereofsomefinitedimensionalsubspace$\mathscr{H}0$of$\mathscr{H}$and$|T|{{{\mathscr{H}}_0}^\perp} < |T|$,thenforevery$S\in$\mathbb{B}$(\mathscr{H})$,$T$isthestrongBirkhoff--Jamesorthogonalto$S$ifandonlyifthereexistsaunitvector$\xi\in {\mathscr{H}}_0$ such that $\mathrm{\parallel }T\mathrm{\parallel }\xi =\mathrm{\mid }T\mathrm{\mid }\xi$ and $S^{\ast }T\xi =0$. Finally, we introduce a new type of approximate orthogonality and investigate this notion in the setting of inner product $C^{\ast }$-modules.