Product of operators and numerical range

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