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# Singular Integral Operators and Essential Commutativity on the Sphere

Published:2010-03-18
Printed: Aug 2010
• Jingbo Xia,
Department of Mathematics, State University of New York at Buffalo, Buffalo, NY 14260, USA
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## Abstract

Let ${\mathcal T}$ be the $C^\ast$-algebra generated by the Toeplitz operators $\{T_\varphi : \varphi \in L^\infty (S,d\sigma )\}$ on the Hardy space $H^2(S)$ of the unit sphere in $\mathbf{C}^n$. It is well known that ${\mathcal T}$ is contained in the essential commutant of $\{T_\varphi : \varphi \in \operatorname{VMO}\cap L^\infty (S,d\sigma )\}$. We show that the essential commutant of $\{T_\varphi : \varphi \in \operatorname{VMO}\cap L^\infty (S,d\sigma )\}$ is strictly larger than ${\mathcal T}$.
 MSC Classifications: 32A55 - Singular integrals 46L05 - General theory of $C^*$-algebras 47L80 - Algebras of specific types of operators (Toeplitz, integral, pseudodifferential, etc.)