Localizations and Essential Commutant of Toeplitz Algebra on Polydisk
Jingming Zhu, Chaohua Zhang

TL;DR
This paper establishes that the norm closure of Toeplitz operators on the polydisk coincides with the Toeplitz algebra, and characterizes its essential commutant, revealing structural properties similar to those in the unit ball case.
Contribution
It proves the equality of the Toeplitz algebra with the norm closure of certain operator families on the polydisk and characterizes the essential commutant of this algebra.
Findings
The norm closure of Toeplitz operators on the polydisk equals the Toeplitz algebra.
The essential commutant of the Toeplitz algebra is characterized by functions of vanishing oscillation plus compact operators.
The Toeplitz algebra satisfies the double commutant relation in the Calkin algebra.
Abstract
Usually, the norm closure of a family of operators is not equal to the -algebra generated by this family of operators. But, similar with the Bergman space of the unit ball in , we show that the norm closure of on Bergman space of the ploydisk in actually coincides with the Toeplitz algebra . A key ingredient in the proof is the class of operators recently introduced by Yi Wang and Jingbo Xia. In fact, as a by-product, we simultaneously proved that also coincides with . Based on these results, we further proved that the essential commutant of Toeplitz algebra equals to where is the collection of…
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Taxonomy
TopicsAdvanced Topics in Algebra · Advanced Algebra and Logic · Algebraic structures and combinatorial models
