Composition operators acting on weighted Dirichlet spaces
type: Item
instance of: scholarly article
title: Composition operators acting on weighted Dirichlet spaces
zbMATH Open document ID: 1293.47022
author: Jordi Pau
author: Patricio A. Pérez
published in: Journal of Mathematical Analysis and Applications
publication date: 2013-04-22T00:00:00Z
review text: The paper is devoted to study composition operators acting on weighted Dirichlet spaces. Given an analytic function on the open unit disk in the complex plane \(\varphi: \mathbb{D}\rightarrow \mathbb{D}\), the associated composition operator \(C_{\varphi}\) acting on the space of holomorphic functions on the disk \(H(\mathbb{D})\) is defined by \(C_{\varphi}f(z)=f(\varphi(z))\). It is an interesting problem to describe the operator properties of \(C_{\varphi}\) in terms of the symbol \(\varphi\) when the operator acts on several spaces of analytic functions in \(\mathbb{D}\). They are well known for the Hardy and Bergman spaces. In the present paper, the authors focus on the weighted Dirichlet spaces \(\mathfrak{D}_{\alpha}\), \(0<\alpha<1\). They consist of those analytic functions \(f\) on \(\mathbb{D}\) with \[ \|f\|_{\mathfrak{D}_{\alpha}}={\left({|f(0)|}^2+\int_{\mathbb{D}}{|f'(z)|}^2\,dA_{\alpha}(z)\right)}^{1/2}<\infty, \] where \(dA_{\alpha}(z)=(1+\alpha)(1-|z|^2)^{\alpha}\,dA(z)\), and \(dA(z)=\frac{1}{\pi}\,dx\,dy\) is the normalized area measure on \(\mathbb{D}\). The continuity of the operator acting on these spaces was characterized in [\textit{K. Kellay} and \textit{P. Lefèvre}, J. Math. Anal. Appl. 386, No. 2, 718--727 (2012; Zbl 1231.47024)] and [\textit{N. Zorboska}, Proc. Am. Math. Soc. 126, No. 7, 2013--2023 (1998; Zbl 0894.47023)]. The authors estimate the essential norm of \(C_{\varphi}\) acting on \(\mathfrak{D}_{\alpha}\), i.e., the distance to the compact operators. They obtain a description of the membership in the Schatten-von~Neumann ideal \(S_p\) of \(\mathfrak{D}_{\alpha}\), that is, the compact operators \(C_{\varphi}\) such that its sequence of singular numbers is in the sequence space \(\ell_p\). Recall that the singular numbers of a compact operator \(T\) are the square root of the eigenvalues of the positive operator \(T^*T\), where \(T^*\) denotes the adjoint of \(T\). This description is done using a relationship of composition op…
Mathematics Subject Classification ID: 47B33
Mathematics Subject Classification ID: 31C25
zbMATH DE Number: 6156276
zbMATH Keywords: composition operators
zbMATH Keywords: Dirichlet spaces
zbMATH Keywords: Schatten classes
reviewed by: María J. Beltrán
MaRDI profile type: Publication
full work available at URL: j.jmaa.2012.12.052
OpenAlex ID: W1993264104
DOI: 10.1016/J.JMAA.2012.12.052
description: scientific article; zbMATH DE number 6156276