Orthogonality of bounded linear operators

Abstract

In this paper we explore the connection between strict convexity of a real normed linear space X and orthogonality of operators in the sense of Birkhoff–James in K(X), the space of all compact linear operators on X. We prove that a real reflexive Banach space X is strictly convex iff for any T,A∈K(X), T⊥BA⇒T⊥SBA or Ax=0 for some x∈SX with ‖Tx‖=‖T‖. We prove that if H is an infinite dimensional real Hilbert space and T∈K(H), then for all A∈B(H), A⊥BT⇒T⊥BA if and only if T is the zero operator. We also prove that for a real Hilbert space H, T⊥BA⇒A⊥BT for all A∈B(H) if and only if T is the zero operator.