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 B (H). Moreover, we obtain a kind of Pythagorean relation for bounded linear operators. In addition, for T is an element of B (H) we prove that if the norm attaining set M-T is a unit sphere of some finite dimensional subspace H-0 of H and vertical bar vertical bar T vertical bar vertical bar (H0 perpendicular to) < vertical bar vertical bar T vertical bar vertical bar, then for every S is an element of B(H), T is the strong Birkhoff-James orthogonal to S if and only if there exists a unit vector xi is an element of H-0 such that vertical bar vertical bar T vertical bar vertical bar xi = vertical bar vertical bar T vertical bar vertical bar 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.