Reducing Subspaces on the Annulus

TL;DR

This paper investigates the structure of reducing subspaces for multiplication operators on the Bergman and Hardy spaces of an annulus, revealing a precise count and extending results to weighted shifts with geometric insights.

Contribution

It establishes that the multiplication operator M_{z^{n}} has exactly 2^n reducing subspaces on the annulus, a novel result contrasting with the disk case, and extends findings to weighted shifts.

Findings

M_{z^{n}} has exactly 2^n reducing subspaces on the annulus

The same reduction count holds for the Hardy space on the annulus

Results are extended to certain bilateral weighted shifts

Abstract

We study reducing subspaces for an analytic multiplication operator M_{z^{n}} on the Bergman space L_{a}^{2}(A_{r}) of the annulus A_{r}, and we prove that M_{z^{n}} has exactly 2^n reducing subspaces. Furthermore, in contrast to what happens for the disk, the same is true for the Hardy space on the annulus. Finally, we extend the results to certain bilateral weighted shifts, and interpret the results in the context of complex geometry.