On the $C^{*}$-Algebra Generated by the Bergman Operator, Carleman Second-Order Shift, and Piecewise Continuous Coefficients

Abstract

We study the $C^{*}$ -algebra generated by the Bergman operator with piecewise continuous coefficients in the Hilbert space $L_2$ and extended by the Carleman rotation by an angle $π$. As a result, we obtain an efficient criterion for the operators from the indicated $C^{*}$ -algebra to be Fredholm operators.