Communications in Mathematical Analysis

Toeplitz Operators with Piecewise Quasicontinuous Symbols

B. Ocampo

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Abstract

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.

Article information

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

Dates
First available in Project Euclid: 18 December 2014

Permanent link to this document
https://projecteuclid.org/euclid.cma/1418919770

Mathematical Reviews number (MathSciNet)
MR3292974

Zentralblatt MATH identifier
1329.30025

Subjects
Primary: 32A36: Bergman spaces 32A40: Boundary behavior of holomorphic functions 32C15: Complex spaces 47B38: Operators on function spaces (general) 47L80: Algebras of specific types of operators (Toeplitz, integral, pseudodifferential, etc.)

Keywords
Bergman spaces C*-algebras Toeplitz operator quasicontinuous symbols piecewise continuous symbols

Citation

Ocampo, B. Toeplitz Operators with Piecewise Quasicontinuous Symbols. Commun. Math. Anal. 17 (2014), no. 2, 263--278. https://projecteuclid.org/euclid.cma/1418919770


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