Compact and quasinormal composition operators

Abstract:

Let ${C_\phi }$ be a composition operator on ${L^2}(\lambda )$, where $\lambda$ is a $\sigma$-finite measure on a set $X$. If $X$ is nonatomic, then Ridge proved that no one-to-one composition operator ${C_\phi }$, with dense range is compact. This result is generalized in the paper by removing one-to-one and dense range conditions. The quasinormal composition operators are also characterized in terms of commutativity with the multiplication operator induced by the Radon-Nikodym derivative of the measure $\lambda {\phi ^{ - 1}}$ with respect to $\lambda$.