Characterization of quasi-parabolic operators and their integral representation

The aim of the paper is to characterize all quasi-parabolic operators and provide an integral representation to each quasi-parabolic operator on the Bergman space \(A_{\lambda }^2(D_n)\). We explore some aspects of operator theoretic properties such as compactness, spectrum, common invariant subspaces and more. Further, we show that the collection of all quasi-parabolic operators forms a maximal commutative \(C^*\)-algebra. As a consequence, we provide integral representation for operators in the \(C^*\)-algebra generated by Toeplitz operators with essentially bounded quasi-parabolic defining symbols.