Commuter of operators in a Hilbert space

Abstract

Commutativity between two stationary functions is a notion that generalizes the concept of stationary correlation. Two stationary functions commute if and only if their associated spectral measures and unitary operators commute. When the operators do not commute, we define the concept of commuter for two operators, and then derive it for two projectors, for a projector and a unitary operator, and for two unitary operators. We establish relations between these commuters and some other tools related to proximity between processes. Finally, we propose a method which retrieves the commuter for two stationary series of finite spectrum, and we study some convergence properties.