## Illinois Journal of Mathematics

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

### Aleksandrov operators as smoothing operators

#### 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