Invertible composition operators on 𝐿²(𝜆)

Abstract

Let C ϕ {C_\phi } be a composition operator on L 2 ( λ ) {L^2}(\lambda ) , where λ \lambda is a σ \sigma -finite measure defined on the Borel subsets of a standard Borel space. In this paper a necessary and sufficient condition for the invertibility of C ϕ {C_\phi } is given in terms of invertibility of ϕ \phi . Also all invertible composition operators on L 2 ( R ) {L^2}({\mathbf {R}}) induced by monotone continuous functions are characterised.