On the essential commutant of analytic Toeplitz operators associated with spherical isometries
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. Bull. 42, No.~7, 548--552 (1997; Zbl 0897.47017)].
scientific article; zbMATH DE number 5925495
scientific article