Product of operators and numerical range

Abstract

We show that a bounded linear operator A is an element of B(H) is a multiple of a unitary operator if and only if AZ and Z A always have the same numerical radius or the same numerical range for all (rank one) Z is an element of B(H). More generally, for any bounded linear operators A, B is an element of B(H), we show that AZ and ZB always have the same numerical radius (resp., the same numerical range) for all (rank one) Z is an element of B(H) if and only if A = e(it) B (resp., A = B) is a multiple of a unitary operator for some t is an element of [0, 2 pi). We extend the result to other types of generalized numerical ranges including the k-numerical range and the higher rank numerical range.