Resolvent estimates and decomposable extensions of generalized Cesàro operators
type: Item
instance of: scholarly article
title: Resolvent estimates and decomposable extensions of generalized Cesàro operators
zbMATH Open document ID: 1185.47031
author: Alexandru Aleman
author: Anna-Maria Persson
published in: Journal of Functional Analysis
publication date: 2010-01-11T00:00:00Z
review text: Let \(H(\mathbb D)\) be the space of all holomorphic functions in the unit disk \(\mathbb D\). For \(g\in H(\mathbb D)\), the generalized Cesàro operator \({\mathfrak C}_g\) is defined by \[ {\mathfrak C}_g f(z)=\frac{1}{z}\int_0^z f(w)g'(w) \,dw, \quad z\in\mathbb D,\;f\in H(\mathbb D). \] The authors determine the spectrum of generalized Cesàro operators for special symbols on various spaces of analytic functions on \(\mathbb D\), including the Hardy spaces \(H^p\), the standard weighted Bergman spaces \[ A^{p,\alpha}= \bigg\{f\in H(\mathbb D): \int_{\mathbb D} |f(z)|^p(1-|z|)^\alpha \,d\sigma_2(z)<\infty\bigg\}, \] the weighted Dirichlet spaces \[ D^{p,\alpha}=\{f\in H(\mathbb D): f '\in A^{p,\alpha}\}, \] as well as the growth classes \[ A^{-\gamma}= \bigg\{f\in H(\mathbb D): \sup_{z\in\mathbb D}(1-|z|)^{\gamma}|f(z)|<\infty\bigg\} \] and its closed subspace \[ A^{-\gamma_0}= \bigg\{f\in A^{-\gamma}: \limsup_{|z|\to 1}(1-|z|)^{\gamma}|f(z)|=0\bigg\}. \] Results on boundedness and compactness of these operators as well as certain weighted composition operators are included, too. Conditions are given under which certain types of these Cesàro operators are subdecomposable. The paper contains a great variety of other interesting material.
reviewed by: Raymond Mortini
Mathematics Subject Classification ID: 47B38
Mathematics Subject Classification ID: 47B40
Mathematics Subject Classification ID: 46E15
zbMATH DE Number: 5655670
zbMATH Keywords: Cesàro operator
zbMATH Keywords: decomposable operators
zbMATH Keywords: Dirichlet space
zbMATH Keywords: Hardy space
zbMATH Keywords: weighted Bergman space
zbMATH Keywords: spectrum
zbMATH Keywords: resolvents
MaRDI profile type: Publication
full work available at URL: j.jfa.2009.10.006
OpenAlex ID: W2083618738
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DOI: 10.1016/J.JFA.2009.10.006
description: scientific article; zbMATH DE number 5655670