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Journal of Operator Theory

Volume 86, Issue 1, Summer 2021  pp. 125-143.

Essential commutants and characterizations of the Toeplitz algebra

Authors:  Raffael Hagger
Author institution: Department of Mathematics and Statistics, University of Reading, Whiteknights Campus, Reading Rg6 6AX, U.K.

Summary: We study the Toeplitz algebra which is generated by Toeplitz operators with bounded symbols on the Fock space $F^p_{\alpha}$. We show that the Toeplitz algebra coincides with each of the algebras generated by band-dominated, sufficiently localized and weakly localized operators, respectively. Moreover, we determine its essential commutant and its essential bicommutant. For $p = 2$ these results were obtained recently by Xia. However, Xia's ideas are mostly connected to Hilbert space theory and methods which are not applicable for $p \neq 2$. Instead, we use a recent result of Fulsche to generalize Xia's theorems.

Keywords: Toeplitz algebra, band-dominated operators, localized operators, Fock space, essential commutants

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