Spectral properties of generalized Cesàro operators
type: Item
instance of: scholarly article
title: Spectral properties of generalized Cesàro operators
zbMATH Open document ID: 1075.47030
author: Scott W. Young
published in: Integral Equations and Operator Theory
publication date: 2004-11-05T00:00:00Z
review text: Let \(g\) be an analytic function on the (open) unit disk \(D\). The generalized Cesàro operator on \({\mathcal H}^2(D)\) with symbol \(g\) is defined by \[ {\mathcal C}_g f (z)=\frac{1}{z}\int_0^z f(t) g(t) \,dt,\quad z\in D. \] For \(g(t)=\frac{1}{1-t}\), \(t\in D\), we obtain the so-called Cesàro operator on \({\mathcal H}^2(D)\). The paper is devoted to the spectral properties of a bounded generalized Cesàro operator \({\mathcal C}_g\) with a rational symbol \(g\) (the spectrum of the Cesàro operator on \({\mathcal H}^2(D)\) has been known for a long time). This means that \(g\) can have only simple poles on the unit circle \(\mathbb T\). The main result of the present paper (Theorem 3.3) characterizes the spectrum of \({\mathcal C}_g\) in terms of the partial fraction decomposition of \(g\) relative to \(\mathbb T\), \[ g(z)=\sum_{i=1}^n \frac{a_i}{1-\beta_i z} + S(z), \] where \(S\) is a rational function analytic in the closed unit disk: namely, the spectrum of \({\mathcal C}_g\) consists of the union of the \(n\) closed disks \(\overline{D (a_i)}\), \(D (a_i)=\{ z: | z-a_i| <| a_i| \), with some possibly isolated points, the set of which is given by \({\mathcal E}=\{ \frac{g(0)}{k} \}_{k=1}^\infty \setminus \cup_{i=1}^n \overline{D (a_i)}\). The essential spectum, \(\sigma_e ({\mathcal C}_g)\), is the union of the \(n\) circles \(\partial D (a_i)\), \(i=1,\ldots, n\), and, moreover, for \(\lambda \notin \sigma_e ({\mathcal C}_g)\), \(\text{ind}\, ({\mathcal C}_g-\lambda I)=-G(\lambda)\), where \(G=\sum_{i=1}^n \chi_{D (a_i)}\). The author uses this result in connection with the BDF theory to show that a generalized Cesàro operator with a rational symbol is unitarily equivalent to a direct sum of multiples of the Cesàro operator, up to a compact perturbation. The case where the poles of the rational function are rationally related is considered in detail. At the end of the paper, some examples are provided, ilustrating cases where the set \({\mathcal E}\) has…
Mathematics Subject Classification ID: 47G10
Mathematics Subject Classification ID: 47A10
Mathematics Subject Classification ID: 32A35
zbMATH DE Number: 2113889
zbMATH Keywords: Cesàro operator
zbMATH Keywords: spectrum
zbMATH Keywords: BDF theory
reviewed by: A. B. Lebre
MaRDI profile type: Publication
full work available at URL: s00020-003-1286-0
OpenAlex ID: W2062808397
DOI: 10.1007/S00020-003-1286-0
description: scientific article; zbMATH DE number 2113889