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

Abstract

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 nontrivial realizations of such commutation relations are given.

9 pages. Material reorganised, new examples added to highlight relations between main results. Submitted to Proc. Roy. Soc. A