Subnormality and composition operators on the Bergman space.
type: Item
instance of: scholarly article
title: Subnormality and composition operators on the Bergman space.
zbMATH Open document ID: 1041.47010
DOI: 10.1007/BF02789595
author: Alexander E. Richman
published in: Integral Equations and Operator Theory
publication date: 2004-01-19T00:00:00Z
review text: \textit{C. Cowen} and \textit{T. Kriete} [J. Funct. Anal. 81, 298--319 (1988; Zbl 0669.47012)] characterized the analytic maps \(\varphi: \mathbb{D}\to \mathbb{D}\) (\(\mathbb{D}=\) the unit disk) for which the composition operator \(C_\varphi \) defined by \(C_\varphi f =f \circ \varphi \) on the Hardy space \(H^2(\mathbb{D})\) has a subnormal adjoint. Specifically, they showed that under a regularity condition near the Denjoy-Wolff point \(d\), \(| d| =1\), \(C_\varphi^*\) is subnormal on \(H^2\) if and only if \[ \varphi (z) = \frac{(r+s)z+(1-s)d}{r(1-s)\overline{d}z+(1+sr)} \] where \(0< s = \varphi ' (d) < 1 \) and \(0 \leq r \leq 1\). \textit{C. Cowen} [Integral Equations Oper. Theory 15, 167--171 (1992; Zbl 0773.47009)] proved that if for \(\varphi \), \(C_\varphi^*\) is subnormal on \(H^2\), then \(\varphi \) also gives rise to a subnormal operator \(C_\varphi^*\) on \(A^2(\mathbb{D})\). In the paper under review, the author characterizes those analytic selfmaps \(\varphi\) of \(\mathbb{D}\), for which \(C_\varphi^*\) is subnormal on \(A^2(\mathbb{D})\). The conditions are as those in the Hardy space case but with large range for the parameter \(r\); in fact, \(r\) is allowed in \([ - 1/7, 1]\).
Mathematics Subject Classification ID: 47B33
Mathematics Subject Classification ID: 47B32
Mathematics Subject Classification ID: 47B20
zbMATH DE Number: 1890557
zbMATH Keywords: composition operator
zbMATH Keywords: subnormal operator
zbMATH Keywords: Bergman space
zbMATH Keywords: Hardy space
zbMATH Keywords: Denjoy-Wolff point
reviewed by: María del Carmen Calderón-Moreno
MaRDI profile type: Publication
cites work: On the Iteration of Analytic Functions in a Halplane II
cites work: Q4000087
cites work: Iteration and the Solution of Functional Equations for Functions Analytic In the Unit Disk
cites work: Q3222739
cites work: Linear fractional composition operators on \(H^ 2\)
cites work: Transferring subnormality of adjoint composition operators
cites work: Subnormality and composition operators on \(H^ 2\)
cites work: Q4888816
cites work: Q5679336
cites work: Q5827353
cites work: Q3027716
cites work: Q4024274
cites work: Q3756035
cites work: Subnormality and Weighted Shifts
cites work: Q5690460
cites work: Q5667036
cites work: Q4340161
full work available at URL: https://doi.org/10.1007/bf02789595
OpenAlex ID: W2059320459
description: scientific article; zbMATH DE number 1890557