A note on commutativity up to a factor of bounded operators

Abstract

In this note, we explore commutativity up to a factor AB = λBA for bounded operators A and B in a complex Hilbert space. Conditions on possible values of the factor A are formulated and shown to depend on spectral properties of the operators. Commutativity up to a unitary factor is considered. In some cases, we obtain some properties of the solution space of the operator equation AX = λxA and explore the structures of A and B that satisfy AB = λBA for some λ ∈ C {0}. A quantum effect is an operator A on a complex Hilbert space that satisfies 0 ≤ A ≤ I. The sequential product of quantum effects A and B is defined by A o B = A 1 /2 BA 1 /2. We also obtain properties of the sequential product.