Communications in Mathematical Analysis

Toeplitz Operators with Piecewise Quasicontinuous Symbols

B. Ocampo

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text


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

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

First available in Project Euclid: 18 December 2014

Permanent link to this document

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

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.)

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


Ocampo, B. Toeplitz Operators with Piecewise Quasicontinuous Symbols. Commun. Math. Anal. 17 (2014), no. 2, 263--278.

Export citation