A Beurling-type theorem in Bergman spaces

It is known that Beurling’s theorem concerning invariant subspaces is not true in the Bergman space (in contrast to the Hardy space case). However, Aleman, Richter, and Sundberg proved that every cyclic invariant subspace in the Bergman space L p a(D), 0 p +∞, is generated by its extremal function (see [3]). This implies, in particular, that for every zero-based invariant subspace in the Bergman space the Beurling’s theorem stands true. Here, we shall supply an alternative proof for this latter statement; our short proof is more direct and closely related to Hedenmalm’s original approach to the problem.