Breitner Ocampo

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 stands for the -algebra of quasicontinuous functions on
$\partial{\mathbb{D}}$ defined by D. Sarason in [5]. We define then as the -algebra generated by and .

$A^2(\mathbb{D})$ 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

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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