On the Reducibility of Weighted Composition Operators Generated by Periodic Transformations

Abstract

In this article, we study the reducibility of weighted composition operators (also known as weighted displacement operators) acting on Banach spaces of continuous functions on a compact topological space X . We consider operators of the form B u ( x ) = a ( x ) u ( α ( x )), where α : X ⟶ X is a continuous mapping and a is a continuous function. The main objective is to determine when such an operator can be reduced, via a Lyapunov transformation (multiplication by an invertible continuous function), to a constant‐coefficient or invariant operator. We establish a link between this reducibility problem and the solvability of a homological equation associated with α . Using the representation theory of the cyclic group for periodic mappings α , we provide conditions for reducibility in terms of algebraic properties of the operator and topological invariants such as the Cauchy index. Examples are given to illustrate the topological obstacles to reducibility.