We consider holomorphic functions 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 ¡p 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

more »

... d by the condition ^ |ûn|2 exp(^/ñ) < oo. The property of Möbiusinvariance plays a crucial and mysterious role in these matters.