A remark on orthogonality and symmetry of operators in B(H)

If T and A are operators on a Hilbert space such that ‖T+λA‖≥‖T‖ for all real numbers λ, then T is said to be r-orthogonal to A. We obtain necessary and sufficient conditions for this to be the case. As a consequence we characterize r-left symmetric (left symmetric) and r-right symmetric (right symmetric) operators in B(H), where H is a real or complex Hilbert space.