Characterizations of Operator Birkhoff–James Orthogonality

Abstract

AbstractIn this paper, we obtain some characterizations of the (strong) Birkhoff–James orthogonality for elements of Hilbert C*-modules and certain elements of . Moreover, we obtain a kind of Pythagorean relation for bounded linear operators. In addition, for we prove that if the norm attaining set is a unit sphere of some finite dimensional subspace of and , then for every , T is the strong Birkhoff–James orthogonal to S if and only if there exists a unit vector such that . Finally, we introduce a new type of approximate orthogonality and investigate this notion in the setting of inner product C*-modules.