Illinois Journal of Mathematics

Aleksandrov operators as smoothing operators

Alec L. Matheson

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A holomorphic function $b$ mapping the unit disk $\disk$ into itself induces a family of measures $\tau_\alpha$, $|\alpha|=1$, on the unit circle $\circle$ by means of Herglotz's Theorem. This family of measures defines the Aleksandrov operator $A_b$ by means of the formula $A_b f(\alpha) = \int_\circle f(\zeta)\,d\tau_\alpha(\zeta)$, at least for continuous $f$. This operator preserves the smoothness classes determined by regular majorants, and is seen to be compact on these classes precisely when none of the measures $\tau_\alpha$ has an atomic part. In the process, a duality theorem for smoothness classes is proved, improving a result of Shields and Williams, and various theorems about composition operators on weighted Bergman spaces are extended to spaces arising from regular weights.

Article information

Illinois J. Math., Volume 45, Number 3 (2001), 981-998.

First available in Project Euclid: 13 November 2009

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Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 47B38: Operators on function spaces (general)
Secondary: 30D45: Bloch functions, normal functions, normal families 30D50


Matheson, Alec L. Aleksandrov operators as smoothing operators. Illinois J. Math. 45 (2001), no. 3, 981--998. doi:10.1215/ijm/1258138164.

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