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Following upon results of Putinar, Sun, Wang, Zheng and the first author, weprovide models for the restrictions of the multiplication by a finite Balschkeproduct on the Bergman space in the unit disc to its reducing subspaces. Themodels involve a generalization of the notion of bundle shift on the Hardyspace introduced by Abrahamse and the first author to the Bergman space. Wedevelop generalized bundle shifts on more general domains. While thecharacterization of the bundle shift is rather explicit, we have not been ableto obtain all the earlier results appeared, in particular, the facts that thenumber of the minimal reducing subspaces equals the number of connectedcomponents 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 Riemannsurface is not made clear also.

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