Operators on the orthogonal complement of the Dirichlet space
type: Item
instance of: scholarly article
title: Operators on the orthogonal complement of the Dirichlet space
zbMATH Open document ID: 1180.47023
author: Tao Yu
published in: Journal of Mathematical Analysis and Applications
publication date: 2009-06-23T00:00:00Z
review text: The purpose of this paper is the study of some properties of dual Hankel and Toeplitz operators on the orthogonal complement of the Dirichlet space of the open unit disk \(\mathbb{D}\). The Sobolev space \(W^{1,2}\) is a Hilbert space of functions with the inner product \[ \langle u,\nu\rangle=\int_{\mathbb{D}}u \,dA\int_{\mathbb{D}}\bar{\nu}\,dA+\left\langle\frac{\partial u}{\partial z},\frac{\partial\nu}{\partial z}\right\rangle_2+\left\langle\frac{\partial u}{\partial\bar{z}}, \frac{\partial \nu}{\partial\bar{z}}\right\rangle_2. \] Here, \(\langle \cdot,\cdot\rangle_2\) represents the inner product in \(L^2(\mathbb{D},dA)\), \(dA\) denotes the normalized Lebesgue measure on \(\mathbb{D}\), and derivatives are considered in the weak sense. The Dirichlet space \(\mathcal{D}\) is the closed subspace of \(W^{1,2}\) consisting of all analytic functions on \(\mathbb{D}\) vanishing at \(0\). Let \(P\) denote the orthogonal projection from \(W^{1,2}\) onto \(\mathcal{D}\). For a function \(\phi\in W^{1,\infty}=\{u\in W^{1,2}:u,\frac{\partial u}{\partial z}, \frac{\partial u}{\partial\bar{z}}\in L^{\infty}(\mathbb{D})\}\), the Toeplitz operator \(T_{\phi}:\mathcal{D}\rightarrow\mathcal{D}\), the Hankel operator \(H_{\phi}:\mathcal{D}\rightarrow\mathcal{D}^{\perp}\), the dual Toeplitz operator \(S_{\phi}:\mathcal{D}^{\perp}\rightarrow \mathcal{D}^{\perp}\) and the dual Hankel operator \(R_{\phi}:\mathcal{D}^{\perp}\rightarrow\mathcal{D}\) are defined, respectively, by \[ T_{\phi}h=P(\phi h), \;H_{\phi}(h)=(I-P)(\phi h), \;S_{\phi}(u)=(I-P)(\phi u) \] and \[ R_{\phi}(u)=P(\phi u), \quad h\in\mathcal{D}, \, u\in\mathcal{D}^{\perp}. \] In the first theorem, the author proves that for \(\phi\in W^{1,\infty}\), \(R_{\phi}=0\) if and only if \(\phi\in\mathcal{D}^{\perp}\). The second theorem deals with commutativity of dual Toeplitz operators. In the Bergman space setting, \textit{K.\,Stroethoff} and \textit{D.--C.\thinspace Zheng} [``Algebraic and spectral properties of dual To…
reviewed by: Željko Čučković
Mathematics Subject Classification ID: 47B35
Mathematics Subject Classification ID: 46E35
zbMATH DE Number: 5568993
zbMATH Keywords: Sobolev space
zbMATH Keywords: Dirichlet space
zbMATH Keywords: dual Toeplitz operator
zbMATH Keywords: dual Hankel operator
MaRDI profile type: Publication
full work available at URL: j.jmaa.2009.04.019
OpenAlex ID: W4212986725
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DOI: 10.1016/J.JMAA.2009.04.019
description: scientific article; zbMATH DE number 5568993