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Toeplitz Operators with Piecewise Quasicontinuous Symbols

Commun. Math. Anal.
Volume 17, Number 2 (2014), 263 - 278

Toeplitz Operators with Piecewise Quasicontinuous Symbols

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In the paper, $QC$ stands for the $C^*$-algebra of quasicontinuous functions on $\partial {\mathbb D}$ defined by D. Sarason in [10]. For a fixed subset $\Lambda:=\{ \lambda_1, \lambda_2, \dots, \lambda_n\}$ of the unit circle $\partial {\mathbb D}$, we define the algebra $PC$ of piecewise continuous functions in $\partial {\mathbb D} \setminus \Lambda$ with one-sided limits at each point $\lambda_k \in \Lambda$. We define $PQC$ as the $C^*$-algebra generated by both $PC$ and $QC$.

$\mathcal{A}^2({\mathbb D})$ stands for the Bergman space of the unit disk ${\mathbb D}$, that is, the space of square integrable and analytic functions defined on ${\mathbb D}$. Let ${\mathcal K} $ denote the ideal of compact operators acting on $\mathcal{A}^2({\mathbb D})$. Our goal is to describe the Calkin algebra ${\mathcal T}_{PQC}/ {\mathcal K}$, where ${\mathcal T}_{PQC}$ is the $C^*$-algebra generated by Toeplitz operators acting on $\mathcal{A}^2({\mathbb D})$ whose symbols are certain extensions of functions in $PQC$. A function defined on $\partial {\mathbb D}$ can be extended to the disk in many ways, the more natural extensions are the harmonic and the radial ones. In the final part of this paper we prove that the description of ${\mathcal T}_{PQC}$ does not depend on the extension chosen.