Commutativity up to a factor of bounded operators in complex Hilbert space

We explore commutativity up to a factor, AB = uBA, for bounded operators in a complex Hilbert space. Conditions on the possible values of the factor u are formulated and shown to depend on spectral properties of the operators involved. Commutativity up to a unitary factor is considered for pairs of self-adjoint operators. Examples of non-trivial realizations of such commutation relations are given