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Reducing subspaces on the annulus. (English) Zbl 1257.47011
Summary: 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.

MSC:
47A15 Invariant subspaces of linear operators
47B37 Linear operators on special spaces (weighted shifts, operators on sequence spaces, etc.)
47B38 Linear operators on function spaces (general)
51D25 Lattices of subspaces and geometric closure systems
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