Invertible composition operators on $L^{2}(\lambda )$

Abstract:

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