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Singular Integral Operators and Essential Commutativity on the Sphere

Published online by Cambridge University Press:  20 November 2018

Jingbo Xia
Jingbo Xia*
Affiliation:
Department of Mathematics, State University of New York at Buffalo, Buffalo, NY 14260, USA, e-mail: jxia@acsu.buffalo.edu
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Abstract

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Let $\mathcal{T}$ be the ${{C}^{*}}$-algebra generated by the Toeplitz operators $\left\{ {{T}_{\varphi }}:\varphi \in {{L}^{\infty }}\left( S,d\sigma \right) \right\}$ on the Hardy space ${{H}^{2}}\left( S \right)$ of the unit sphere in ${{C}^{n}}$. It is well known that $\mathcal{T}$ is contained in the essential commutant of $\left\{ {{T}_{\varphi }}:\varphi \in \text{VMO}\cap {{L}^{\infty }}\left( S,d\sigma \right) \right\}$. We show that the essential commutant of $\left\{ {{T}_{\varphi }}:\varphi \in \text{VMO}\cap {{L}^{\infty }}\left( S,d\sigma \right) \right\}$ is strictly larger than $\mathcal{T}$.

Keywords

32A5546L0547L80
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 62 , Issue 4 , 01 August 2010 , pp. 889 - 913
Copyright
Copyright © Canadian Mathematical Society 2010

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