Compactness and Univalence

Abstract

We are now ready to classify the univalent self-maps of U that induce compact composition operators on H ². A fragment of operator theoretic folk-wisdom will help us guess the answer:

If a “big-oh” condition describes a class of bounded operators, then the corresponding “little-oh” condition picks out the subclass of compact operators.

(Problem 1 of §2.6 shows this principle in action). Unfortunately, no hint of any “big-oh” condition emerged from our proof of Littlewood’s Theorem, so in this chapter the first order of business has to be:

Find the “right” proof of boundedness for composition operators acting on H ².

Given what we have already done, it should come as no surprise that everything will depend on getting the “right” representation of the H ² norm.