2015): Product of operators and numerical range, Linear and Multilinear Algebra, We show that a bounded linear operator A ∈ 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 ∈ B(H ). More generally, for any bounded linear operators A, B ∈ B(H ), we show that AZ and Z B always have the same numerical radius (resp., the same numerical range) for all (rank one) Z ∈ B(H ) if and only if A = e it B (resp., A = B) is a multiple of a unitary operator for some t ∈ [0, 2π). We extend the result to other types of generalized numerical ranges including the k-numerical range and the higher rank numerical range.