Zero-based invariant subspaces in the Bergman space

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 Sundberge proved that every cyclic invariant subspace in the Bergman space La(D), 0 < p < +∞, is generated by its extremal function. This implies, in particular, that for every zero-based invariant subspace in the Bergman space the Beurling’s theorem stands true. Here, we calculate the reproducing kernel of the zero-based invariant subspace Mn in the Bergman space La(D) where the associated wandering subspaceMn zMn is one-dimensional, and spanned by the unit vector Gn(z) = √ n+ 1zn.