Commutativity up to a factor of bounded operators in complex Hilbert   space
  We explore commutativity up to a factor, $AB=\lambda BA$, for bounded operators in a complex Hilbert space. Conditions on the possible values of the factor $\lambda$ 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 nontrivial realizations of such commutation relations are given. 
J.A. Brooke, P. Busch, D.B. Pearson