Breitner Ocampo (Vol. 19 No. 1 2015)

Toeplitz operators with piecewise quasicontinuous symbols   

Breitner Ocampo

For a fixed subset of the unit circle \partial{\mathbb{D}}, \Lambda : = \{\lambda_1, \lambda_2, ... \lambda_n\}, we define the algebra P C of piecewise continuous functions in
\partial{\mathbb{D}} \textbackslash \Lambda with one sided limits at each point \lambda_k \in \Lambda. Besides, we let QC stands for the  C^*-algebra of quasicontinuous functions on
\partial{\mathbb{D}} defined by D. Sarason in [5]. We define then PQC as the C^*-algebra generated by PCand QC.

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}. Our goal is to describe T_{PQC}, the algebra generated by Toeplitz operators whose symbols are certain extensions of functions in 
PQC acting on A^2(\mathbb{D}). Of course, 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 paper we describe the algebra T_{PQC} and we prove that this description does not depend on the extension chosen.

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