Compact composition operators on spaces of boundary-regular holomorphic functions

Abstract:

We consider holomorphic functions $\phi$ taking the unit disc $U$ into itself, and Banach spaces $X$ consisting of functions holomorphic in $U$ and continuous on its closure; and show that under some natural hypotheses on $X$: if $\phi$ induces a compact composition operator on $X$, then $\phi (U)$ must be a relatively compact subset of $U$. Spaces $X$ which satisfy the hypotheses of this theorem include the disc algebra, "heavily" weighted Dirichlet spaces, spaces of holomorphic Lipschitz functions, and the space of functions with derivative in a Hardy space ${H^p}(p \geq 1)$. It is well known that the theorem is not true for "large" spaces such as the Hardy and Bergman spaces. Surprisingly, it also fails in "very small spaces," such as the Hilbert space of holomorphic functions $f(z) = \sum {{a_n}{z^n}}$ determined by the condition $\sum {|{a_n}{|^2}\exp \left ( {\sqrt n } \right ) < \infty }$. The property of Möbiusinvariance plays a crucial and mysterious role in these matters.