Product of operators and numerical range

Abstract

We show that a bounded linear operator is a multiple of a unitary operator if and only if and always have the same numerical radius or the same numerical range for all (rank one) . More generally, for any bounded linear operators , we show that and always have the same numerical radius (resp., the same numerical range) for all (rank one) if and only if (resp., ) is a multiple of a unitary operator for some . We extend the result to other types of generalized numerical ranges including the -numerical range and the higher rank numerical range.