On a Banach algebra generated by the Bergman operator, constant coefficients, and finitely generated groups of shifts

Abstract

We study a Banach algebra generated by the Bergman operator, constant coefficients, and shifts formed by finitely generated commutative groups of hyperbolic transformations of the unit disk acting in the Lebesgue space $L_p, p > 1$, and obtain an effective criterion for the operators from the considered Banach algebra to be Fredholm operators.