Spectra of composition operators on the Bloch and Bergman spaces
type: Item
instance of: scholarly article
title: Spectra of composition operators on the Bloch and Bergman spaces
zbMATH Open document ID: 1024.47009
DOI: 10.1007/BF02785430
author: Karen Saxe
author: Barbara D. MacCluer
published in: Israel Journal of Mathematics
publication date: 2003-11-14T00:00:00Z
review text: Let \(U\) be the unit disc in the complex plane and \(\varphi\) be an analytic map from \(U\) to itself. In [J. Am. Math. Soc. 10, 299-325 (1997; Zbl 0870.30018)], \textit{P. S. Bourdon} and \textit{J. H. Shapiro} give the essential spectral radius formula on the Hardy space \(H^p\), for any analytic \(\varphi\) and \(0<p<\infty\). Using the Calderón theory of complex interpolation, the authors obtain an analogous result on the Bergman space \(A^p\), namely that for \(1<p<\infty\), \[ \bigl( r_{e,A^p}(C_\varphi) \bigr)^p = \bigl( r_{e,A^2}(C_\varphi) \bigr)^2. \] They also obtain the spectrum of the composition operator \(C_\varphi\) on the Bergman space, Bloch space, BMOA and Hardy space for \(\varphi\) univalent, not an automorphism, with fixed point in \(U\).
Mathematics Subject Classification ID: 47B33
Mathematics Subject Classification ID: 32A35
Mathematics Subject Classification ID: 32A36
zbMATH DE Number: 1819992
zbMATH Keywords: spectrum
zbMATH Keywords: composition operator
zbMATH Keywords: Bloch space
zbMATH Keywords: Bergman space
reviewed by: Jinkee Lee
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1819992