Putnam's theorem, Alexander's spectral area estimate, and VMO
This paper proves that if f is a bounded analytic function defined on the unit disk such that at each point of the unit circle the cluster set of f has area zero, then f is in the space VMO (vanishing mean oscillation). This result and its quantitative version are proved by using techniques from operator theory, especially Putnam's theorem. The above result is extended to the unit ball in \({\mathbb{C}}^ n\), using commutative Banach algebra techniques rather than operator theory. Here the key tool is Alexander's spectral area estimate. Finally, the operator theoretic and function theoretic techniques are tied together by giving a simple proof of Putnam's theorem for subnormal operators. Here the main tool is a quantitative version of the Hartogs-Rosenthal theorem.
scientific article; zbMATH DE number 3859465
scientific article