ZERO-BASED INVARIANT SUBSPACES IN THE BERGMAN SPACE

Abstract

It is known that Beurlingís 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 L p a(D), 0 < p < +1, is generated by its extremal function. This implies, in particular, that for every zero-based invariant subspace in the Bergman space the Beurlingís theorem stands true. Here, we calculate the reproducing kernel of the zero-based invariant subspace Mn in the Bergman space L2 a(D) where the associated wandering subspace Mn zMn is one-dimensional, and spanned by the unit vector Gn(z) = p n + 1z n.