On the essential commutant of analytic Toeplitz operators associated with spherical isometries
type: Item
instance of: scholarly article
title: On the essential commutant of analytic Toeplitz operators associated with spherical isometries
zbMATH Open document ID: 1236.47022
author: Michael Didas
author: Jörg Eschmeier
author: Kevin Everard
published in: Journal of Functional Analysis
publication date: 2011-07-18T00:00:00Z
review text: To state the main result of this interesting paper requires a little background and notation. Recall that a spherical isometry on a complex Hilbert space \(H\) is a commuting tuple \(T\in B(H)^n\) that satisfies \[ \sum_{j=1}^n T_j^* T_j=I_H. \] Also associated to \(T\), there is an abstract theory of \(T\)-Toeplitz operators. These are the operators \(X\in B(H)\) such that \[ \sum_{j=1}^nT_j^*XT_j=X. \] For a general tuple, the point spectrum of the operator is given by \[ \sigma_p(T)=\left\{\xi\in\mathbb{C}^n:\bigcap_{j=1}^n\text{ker}\left(\xi_j-T_j\right)\right\}. \] A tuple \(T\in B(H)^n\) is essentially normal if its self-commutators are compact, namely, \(T_jT_j^*-T_j^*T_j\) is a compact operator on \(H\). Associated to the operator tuple \(T\), one has the dual algebra \[ \mathcal{A}_T=\overline{\mathbb{C}\left[T_1,\dots, T_n\right]}^{w*}\subset B(H). \] Furthermore, the essential commutant of \(\mathcal{A}_T\) is defined by the collection of all \(C\in B(H)\) such that \(CY-YC\) is compact for all \(Y\in \mathcal{A}_T\). The authors then prove that, if \(T\in B(H)^n\) is an essentially normal spherical isometry with empty point specturm, then every operator \(S\) that belongs to the essential commutant of \(\mathcal{A}_T\) has the form \(S=X+K\) with \(X\) a \(T\)-Toeplitz operator and \(K\) a compact operator on \(H\). They further prove that this result holds in the more general setting of \(A\)-isometries (see the paper for the relevant definitions). The authors apply their result to determine the essential commutant of the set of all analytic Toeplitz operators for the Hardy space \(H^2(\sigma)\), where \(\sigma\) is the surface measure on the boundary \(\partial\Omega\) of a strictly pseudoconvex domain \(\Omega\subset\mathbb{C}^n\). This result of the authors extends previous work of \textit{K. R. Davidson} [J. Funct. Anal. 24, 291--302 (1977; Zbl 0343.47022)] for the unit disc and \textit{X.-H. Ding} and \textit{S.-H. Sun} for the unit ball [Chin. Sci…
Mathematics Subject Classification ID: 47B35
Mathematics Subject Classification ID: 47L30
Mathematics Subject Classification ID: 47A13
zbMATH DE Number: 5925495
zbMATH Keywords: spherical isometries
zbMATH Keywords: inner functions
zbMATH Keywords: Toeplitz operators
zbMATH Keywords: essential commutant
zbMATH Keywords: Hardy spaces
reviewed by: Brett Wick
MaRDI profile type: Publication
full work available at URL: j.jfa.2011.05.005
OpenAlex ID: W2000771250
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DOI: 10.1016/J.JFA.2011.05.005
description: scientific article; zbMATH DE number 5925495