Illinois Journal of Mathematics

Aleksandrov operators as smoothing operators

Alec L. Matheson

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Abstract

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

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

Dates
First available in Project Euclid: 13 November 2009

Permanent link to this document
https://projecteuclid.org/euclid.ijm/1258138164

Digital Object Identifier
doi:10.1215/ijm/1258138164

Mathematical Reviews number (MathSciNet)
MR1879248

Zentralblatt MATH identifier
0994.47031

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

Citation

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


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