Compact composition operators on the Smirnov class

Abstract

We show that a composition operator on the Smirnov class N + N^+ is compact if and only if it is compact on some (equivalently: every) Hardy space H p H^p for 0 > p > ∞ 0>p>\infty . Along the way we show that for composition operators on N + N^+ both the formally weaker notion of boundedness, and a formally stronger notion we call metric compactness, are equivalent to compactness.