Journal of Operator Theory

Volume 75, Issue 1, Winter 2016 pp. 3-19.

Generalized bundle shift with application to multiplication operator on the Bergman space

Authors: Ronald G. Douglas (1), Dinesh Kumar Keshari (2), and Anjian Xu (3)
Author institution: (1) Department of Mathematics, Texas A&M University, College Station, TX, 77843, U.S.A.
(2) Department of Mathematics, Texas A&M University, College Station, TX, 77843, U.S.A.
(3) Department of Mathematics, Chongqing University of Technology, Chongqing, China, 400054, and Department of Mathematics, Texas A&M University, College Station, TX, 77843, U.S.A.

Summary: Following upon results of Putinar, Sun, Wang, Zheng and the first author, we provide models for the restrictions of the multiplication by a finite Blaschke 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 also not made clear.

DOI: http://dx.doi.org/10.7900/jot.2014sep05.2051
Keywords: Bergman bundle shift, Bergman space, finite Blaschke product, reducing subspace

Contents Full-Text PDF