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

James A. Brooke, Paul Busch, David B. Pearson


We explore commutativity up to a factor, AB = λBA, for bounded operators in a complex Hilbert space. Conditions on the possible values of the factor λ 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.