An elementary introduction to Clark measures.(English)Zbl 1148.47001
Girela Álvarez, Daniel (ed.) et al., Topics in complex analysis and operator theory. Proceedings of the winter school in complex analysis and operator theory, Malaga, Spain, February 5–9, 2006. Malaga: Universidad de Malaga (ISBN 978-84-9747-174-9/pbk). 85-136 (2007).
For an analytic self-map \(\varphi\) of \(\mathbb D\) and a point \(\alpha \in \mathbb T\), the function \(\frac{\alpha -\varphi}{\alpha + \varphi}\) is also analytic on \(\mathbb D\) and has a positive real part. Thus, a positive measure \(\mu_{\alpha}\) on \(\mathbb T\) is given by \[ \text{Re}\left ( \frac{\alpha - \varphi(z)}{\alpha + \varphi(z)} \right ) = \int_{\mathbb T} \frac{1-|z|^2}{\xi - z|^2} \; d \mu_{\alpha(\xi)}, \; z \in \mathbb D. \] Measures of this type were introduced byD.N.Clark in [J. Anal.Math.25, 169–191 (1972;Zbl 0252.47010)] and thoroughly studied by A.B.Aleksandrov. In the literature they are known as Clark measures, spectral measures, Aleksandrov measures, or Aleksandrov–Clark measures.
The paper under review is a well written introduction to the concept of Clark measures based on notes of a minicourse held by the author at the conference ‘Winter School in Complex Analysis and Operator Theory’ in Antequera (Málaga), Spain, in February 2006.
The author starts by giving the basic properties of Clark measures and uses these to derive some properties of the corresponding Aleksandrov operator. The adjoint of an Aleksandrov operator is the analytic composition operator. Here, those properties of composition operators which are closely related to Clark measures are discussed. Further topics covered in this article are the boundary value distribution of self-maps of \(\mathbb D\), the original idea of Clark to obtain the spectral resolution of so-called model operators in terms of Clark measures and a generalization of Plancherel’s theorem which is due to A.Poltoratski.
For the entire collection see [Zbl 1128.30004].
The paper under review is a well written introduction to the concept of Clark measures based on notes of a minicourse held by the author at the conference ‘Winter School in Complex Analysis and Operator Theory’ in Antequera (Málaga), Spain, in February 2006.
The author starts by giving the basic properties of Clark measures and uses these to derive some properties of the corresponding Aleksandrov operator. The adjoint of an Aleksandrov operator is the analytic composition operator. Here, those properties of composition operators which are closely related to Clark measures are discussed. Further topics covered in this article are the boundary value distribution of self-maps of \(\mathbb D\), the original idea of Clark to obtain the spectral resolution of so-called model operators in terms of Clark measures and a generalization of Plancherel’s theorem which is due to A.Poltoratski.
For the entire collection see [Zbl 1128.30004].
Reviewer: Elke Wolf (Paderborn)
MSC:
| 47-01 | Introductory exposition (textbooks, tutorial papers, etc.) pertaining to operator theory |
| 30D40 | Cluster sets, prime ends, boundary behavior |
| 30E20 | Integration, integrals of Cauchy type, integral representations of analytic functions in the complex plane |
| 47B33 | Linear composition operators |
| 47B38 | Linear operators on function spaces (general) |
