Colloquium

Complemented invariant subspaces in Bergman spaces and their projectors

by Boris Korenblum, Professor of Mathematics, University at Albany, SUNY

Abstract

Despite recent advances in the Function Theory of Bergman spaces, one fundamental question remains open: Is every z-invariant subspace I of the Bergman space A^p (p not equal to 2) complemented, (i.e. does there exist a bounded projector from A^p to I)? In contrast, for the Hardy spaces H^p, it is well known (at least for 1 < p < ¥) that every invariant subspace I = I_j = jH^p (j inner) generates a projector P_I : H^p ---> I, P_I = jT_y where y is the conjugate of j and T_y is the Toeplitz operator with the symbol y. It turns out that the above formula can be modified to provide projectors from A^p (1 < p < ¥) to I Ì A^p for invariant subspaces I associated with atomic measures.