Toeplitz operators and algebras on Dirichlet spaces.
type: Item
instance of: scholarly article
title: Toeplitz operators and algebras on Dirichlet spaces.
zbMATH Open document ID: 1047.47054
DOI: 10.1142/S0252959902000353
published in: Chinese Annals of Mathematics. Series B
publication date: 2002-01-01T00:00:00Z
review text: Let \(D\subset \mathbb{C}\) be the unit disk, \(dA(z)=\frac{dxdy}{\pi}\) and let \(L^{2,1}\) denote the Sobolev space consisting of functions \(u\) such that \(\int_{D} \left(| \frac{\partial u}{\partial z}| ^2+| \frac{\partial u}{\partial \overline z}| ^2 +| u| ^2\right)dA <\infty\). By \(\mathcal D\) denote the Dirichlet space, that is, the subspace of analytic functions in \(L^{2,1}\). For a symbol \(\varphi \in C^1(D)\), the Toeplitz operator is defined as \(T_a=P(\varphi f), f \in \mathcal D,\) where \(P\) is an orthogonal projection from \(L^{2,1}\) onto \(\mathcal D\). The author studies the automorphism group of the Toeplitz \(C^\ast\)-algebra. He also shows that such Toeplitz operators have connected spectra and describes some algebraic properties of Toeplitz operators with \(C^1\)-symbols.
reviewed by: N. Karapetiants
Mathematics Subject Classification ID: 47L80
Mathematics Subject Classification ID: 47B35
zbMATH DE Number: 1833172
zbMATH Keywords: Toeplitz operator
zbMATH Keywords: Toeplitz algebra
zbMATH Keywords: Dirichlet spaces
author: Guangfu Cao
MaRDI profile type: Publication
full work available at URL: s0252959902000353
OpenAlex ID: W2273711336
description: scientific article; zbMATH DE number 1833172