Putnam's theorem, Alexander's spectral area estimate, and VMO

Abstract

In this paper we show that ] f f is a bounded analytic function defined on the unit disk such that at each point of the unit circle the cluster set o f f has area zero, then f has vanishing mean oscillation (see Sect. 1 for definitions). We discovered this result (Corollary 1.5) and its quantitative version (Theorem 1.4) using some techniques from operator theory. This proof is given in Sect. 1; the main tool is Putnam's Theorem (Lemma 1.1). In the main result in Sect. 2 (Theorem 2.4) we extend the above result to the unit ball in ~ ' , using commutative Banach algebra techniques rather than operator theory. Here the key tool is Alexander's spectral area estimate (Lemma 2.2). Section 3 ties the techniques of Sects. 1 and 2 together by giving a simple proof 0fPutnam's Theorem for subnormal operators. Here the main tool (Lemma 3.2) is a quantitative version of the Hartogs-Rosenthal Theorem. Some concluding remarks indicate another method of combining the tools used in this paper. Although there are obvious connections between the three sections of the paper, each section can be read independently. Throughout the paper, c denotes an absolute constant, independent of everything except the dimension n of ~n (in Sect. 2). However, the actual value of c may change; thus sometimes c 2 is replaced by c.