Commuting dual Toeplitz operators on the orthogonal complement of the Dirichlet space
type: Item
instance of: scholarly article
title: Commuting dual Toeplitz operators on the orthogonal complement of the Dirichlet space
zbMATH Open document ID: 1198.47048
author: Tao Yu
author: Shi Yue Wu
publication date: 2009-11-11T00:00:00Z
review text: Let \(D\) be the open unit disk in the complex plane and let \(dA\) denote the normalized Lebesgue measure. The Sobolev space \(W^{1,2}\) consists of functions \(u:D\to\mathbb{C}\) with the weak partial derivatives of order 1 and the norm \[ \|u\|_{\frac{1}{2}}=\left(\left|\int_D u dA\right|^2+\int_D\left(\left|\frac{\partial u}{\partial z}\right|^2+\left|\frac{\partial u}{\partial\overline{z}}\right|^2\right)dA\right)^{\frac{1}{2}}. \] The Dirichlet space \({\mathcal D}_0\) is the closed subspace of the Hilbert space \(W^{1,2}\) that consists of all holomorphic functions \(f\in W^{1,2}\) with \(f(0)=0\). Let \[ W^{1,\infty}(D)=\left\{\varphi\in W^{1,2}: \varphi,\;\frac{\partial u}{\partial z},\;\frac{\partial u}{\partial\overline{z}}\in L^\infty(D)\right\}. \] Given \(\varphi\in W^{1,\infty}(D)\), the dual Toeplitz operator \(S_\varphi:{\mathcal D}_0^\bot\to{\mathcal D}_0^\bot\) is defined by \(S_\varphi u=Q(\varphi u)\) for \(u\in{\mathcal D}_0^\bot\), where \(Q\) is the orthogonal projection from \(W^{1,2}\) onto \({\mathcal D}_0^\bot\), the orthogonal complement of \({\mathcal D}_0\) in \(W^{1,2}\). Theorem 1: Suppose that \(\varphi,\psi\in W^{1,\infty}(D)\) are harmonic functions. Then \(S_\varphi\) and \(S_\psi\) commute if and only if either both \(\varphi\) and \(\psi\) are holomorphic on \(D\), or both \(\varphi\) and \(\psi\) are antiholomorphic on \(D\), or a nontrivial linear combination of \(\varphi\) and \(\psi\) is constant on \(D\). Theorem 2: Suppose that \(\varphi,\psi\in W^{1,\infty}(D)\) are harmonic functions. Then \(S_\varphi S_\psi=S_{\varphi\psi}\) if and only if either \(\varphi\) is holomorphic on \(D\), or \(\psi\) is antiholomorphic on \(D\). Theorem 3: If \(\varphi,\psi\in W^{1,\infty}(D)\), and \(S_\varphi S_\psi\) is a finite rank perturbation of a dual Toeplitz operator \(S_u\) for some \(u\in W^{1,\infty}(D)\), then \(\varphi\psi=u\).
reviewed by: Yuri I. Karlovich
Mathematics Subject Classification ID: 47B35
zbMATH DE Number: 5629497
zbMATH Keywords: Sobolev space
zbMATH Keywords: Dirichlet space
zbMATH Keywords: orthogonal complement
zbMATH Keywords: dual Toeplitz operator
zbMATH Keywords: commutation
MaRDI profile type: Publication
full work available at URL: s10114-008-7109-0
OpenAlex ID: W2126475011
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DOI: 10.1007/S10114-008-7109-0
published in: Acta Mathematica Sinica, English Series
description: scientific article; zbMATH DE number 5629497