Putnam's theorem, Alexander's
spectral area estimate, and VMO
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Sheldon Axler and Joel H. Shapiro
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Math. Ann. 271
(1985), 161--183.
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| 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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