Fredholm properties of Toeplitz operators on Dirichlet spaces
type: Item
instance of: scholarly article
title: Fredholm properties of Toeplitz operators on Dirichlet spaces
zbMATH Open document ID: 0940.47024
published in: Pacific Journal of Mathematics
publication date: 2000-07-24T00:00:00Z
review text: The author analyzes Toeplitz operators on Dirichlet spaces and computes spectra of Toeplitz operators with symbols in \(C^1,H^\infty_1+ C^1\). Let \(D\) be the unit disk in the complex plane \(\mathbb{C}\), \(dA={1\over \pi} dx dy\) be the normalized area measure on \(D\). \(L^{2,1}\) is the space of functions \(u:D\to\mathbb{C}\), for which the norm \[ \|u\|_{{1\over 2}}= \left[\int_D \Biggl(\Biggl|{\partial u\over\partial z}(z) \Biggr|^2+ \Biggl|{\partial u\over\partial\overline z}(z)\Biggr|^2 dA\Biggr)\right]^{{1\over 2}}<\infty. \] \(L^{2,1}\) is a Hilbert space with the inner product \[ \langle u,v\rangle_{{1\over 2}}= \Biggl\langle{\partial u\over\partial z}, {\partial v\over\partial z}\Biggr\rangle_{L^2(dA)}+ \Biggl\langle{\partial u\over\partial\overline z}, {\partial v\over\partial\overline z}\Biggr\rangle_{L^2(dA)}. \] The Dirichlet space \({\mathbf D}\) is the subspace of all analytic functions \(g\) in \(L^{2,1}\) with \(g(0)= 0\). Let \(P\) be the orthogonal projection from \(L^{2,1}\) into \({\mathbf D}\). \(P\) is an integral operator defined by \[ P(u)(w)= \int_D {\partial u\over\partial z} {\overline{\partial K(z,w)}\over\partial z} dA, \] where \(K(z,w)= \sum^\infty_{k= 1}{z^kw^k\over k}\) is the reproducing kernel of \({\mathbf D}\). Definition 1. Suppose \(\varphi\in C^1(D)\), the operator \[ T_\varphi f(w)= P(\varphi f, K)_{{1\over 2}}= \int_D {\partial(\varphi f)\over\partial z} {\overline{\partial K(z,w)}\over\partial z} dA(z) \] is said to be Toeplitz operator with symbol \(\varphi\). In the first section of the article, the author computes the spectra and essential spectra of Toeplitz operators with symbol \(\varphi\). In the next section, the author proves that \(H^\infty_1+ C^1(D)\) is a norm-closed space relative to a suitable norm, and obtains the representation of essential spectra of the Toeplitz operators with symbols \(H^\infty_1+ C^1(D)\).
Mathematics Subject Classification ID: 47B35
Mathematics Subject Classification ID: 47A53
zbMATH DE Number: 1344303
zbMATH Keywords: Fredholm properties
zbMATH Keywords: Toeplitz operators
zbMATH Keywords: Dirichlet spaces
zbMATH Keywords: essential spectra
author: Guangfu Cao
reviewed by: Andreiy Kondrat'yev
MaRDI profile type: Publication
full work available at URL: pjm.1999.188.209
OpenAlex ID: W2028904767
DOI: 10.2140/PJM.1999.188.209
description: scientific article; zbMATH DE number 1344303