Singular Integral Operators and Essential Commutativity on the Sphere

Xia, Jingbo

Canadian journal of mathematics, Tome 62 (2010) no. 4, pp. 889-913

Résumé

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 $\mathcal{T}$ .

Détail

DOI : 10.4153/CJM-2010-038-x

Mots-clés : 32A55, 46L05, 47L80

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