Composition operator

Summary

A composition operator acts on a space of functions by precomposing with a fixed map: $C_\varphi f = f \circ \varphi$. It is always linear. The questions are when it maps the space into itself, when it is bounded or compact, and how its properties reflect the symbol $\varphi$. On the Hardy space of the disc, every holomorphic self-map $\varphi$ of the disc gives a bounded composition operator, by Littlewood's subordination principle. In ergodic theory the same construction is called the Koopman operator.