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


Sheldon Axler and Joel H. Shapiro


Math. Ann. 271 (1985), 161--183.
Abstract: This paper relates function theory, several complex variables, operator theory and Banach algebras. In the first section a distance estimate is given which shows that if at each point of the boundary of the unit disc D, the cluster set of a bounded analytic function has area zero, then the radial limit function has vanishing mean oscillation. The proof is based on Putnam s theorem on hyponormal operators; it becomes easier for the special class of subnormal operators In the third section we present a proof for this case based on a quantitative version of the Hartogs-Rosenthal theorem from function algebras that is due to H. Alexander. We use Alexander's spectral area estimate to obtain estimates for the BMOA-norms for an analytic function in terms of the area of its image.


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