Putnam's theorem, Alexander's spectral area estimate, and VMO
type: Item
instance of: scholarly article
title: Putnam's theorem, Alexander's spectral area estimate, and VMO
zbMATH Open document ID: 0541.30021
DOI: 10.1007/BF01455985
author: Sheldon Axler
author: Joel H. Shapiro
published in: Mathematische Annalen
publication date: 1985-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/163962
review text: 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.
Mathematics Subject Classification ID: 30D50
Mathematics Subject Classification ID: 32A30
zbMATH DE Number: 3859465
zbMATH Keywords: bounded analytic function
zbMATH Keywords: cluster set
zbMATH Keywords: vanishing mean oscillation
zbMATH Keywords: unit ball
zbMATH Keywords: subnormal operators
zbMATH Keywords: Hartogs-Rosenthal theorem
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 3859465