## Illinois Journal of Mathematics

### Aleksandrov operators as smoothing operators

Alec L. Matheson

#### 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

#### 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