Compact composition operators on spaces of boundary-regular holomorphic functions

We consider holomorphic functions taking the unit disc U into itself, and Banach spaces X consisting of functions holomorphic in U and con- tinuous on its closure; and show that under some natural hypotheses on X: if ip induces a compact composition operator on X, then 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 Hubert space of holomorphic functions f(z) = ^ a"zn determined by the condition ^ |un|2 exp(^/n) (U) C U. Thus induces on H(U), the space of functions holomorphic on U, a composition operator Ctj, defined by the equation C$f — f ° 4> (f G H(U)). There has been growing interest in the study of composition operators on Banach spaces of functions holo- morphic on U, the idea being to connect the behavior of the operator C$ with the function theoretic properties of . The most popular setting for this research has been the Hardy space Hp, but more recently the weighted Bergman and Dirichlet spaces have begun to assert themselves (see (7, 15), for example). Recent work on composition operators in these spaces includes studies of spectra (4), algebras gen- erated by composition operators (3, 9), and compactness of composition operators