Generalized Bundle Shift with Application to Multiplication operator on the Bergman space

TL;DR

This paper extends the concept of bundle shifts to the Bergman space, providing models for multiplication operators restricted to reducing subspaces, and explores their properties on general domains.

Contribution

It generalizes the notion of bundle shifts from Hardy to Bergman spaces and develops models for multiplication operators on these spaces, broadening the theoretical framework.

Findings

Models for restrictions of multiplication by finite Blaschke products on Bergman space.

Generalization of bundle shift concept to more general domains.

Partial characterization of the algebra of commutants and reducing subspaces.

Abstract

Following upon results of Putinar, Sun, Wang, Zheng and the first author, we provide models for the restrictions of the multiplication by a finite Balschke product on the Bergman space in the unit disc to its reducing subspaces. The models involve a generalization of the notion of bundle shift on the Hardy space introduced by Abrahamse and the first author to the Bergman space. We develop generalized bundle shifts on more general domains. While the characterization of the bundle shift is rather explicit, we have not been able to obtain all the earlier results appeared, in particular, the facts that the number of the minimal reducing subspaces equals the number of connected components of the Riemann surface $B(z)=B(w)$ and the algebra of commutant of $T_{B}$ is commutative, are not proved. Moreover, the role of the Riemann surface is not made clear also.