Orthogonality of bounded linear operators

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