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

Abstract

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.

Curvature inequalities and extremal operators

“…It was shown in [14] that the class of operators in F 2 include the bundle shifts introduced in [1]. We conclude this section by showing that the class F 1 includes the class of Bergman bundle shift of rank 1 introduced in [7]. Let G be the class of operators contained in F defined by G = {M on (A 2 (Ω), hdv): log h is harmonic on Ω}.…”

“…After recalling the definition of of Bergman bundle shift (cf. [7]), we proceed to establish the existence of a surjective map from G onto the class of Bergman bundle shift of rank 1.…”

Restriction of Toeplitz Operators on Their Reducing Subspaces

“…In [11], Douglas et al generalized the bundle shift [12] to the case of Bergman spaces, constructed a vector bundle model for analytic Toeplitz operator on the Bergman space 2 (D), and tried to build vector bundle models for restrictions of to its minimal reducing subspaces, but it is not completed. Douglas [13] studied unitary equivalence of the restrictions by computing their curvatures of corresponding geometric models.…”