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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, 46L05, 47L80 show english descriptions Singular integrals
General theory of $C^*$-algebras
Algebras of specific types of operators (Toeplitz, integral, pseudodifferential, etc.)
32A55 - Singular integrals
46L05 - General theory of $C^*$-algebras
47L80 - Algebras of specific types of operators (Toeplitz, integral, pseudodifferential, etc.)
 

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