  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.
