Complemented Invariant Subspaces and Interpolation Sequences

It is shown that the invariant subspace of the Bergman space Ap of the unit disc, generated by a finite union of Hardy interpolation sequences, is complemented in Ap. Let D be the open unit disc of the complex plane C, and let H = H(D) denote the usual Hardy space. For 0 < p < ∞ the Bergman space A consists of analytic functions f on D such that ‖f‖p = ∫ D |f(z)| dA(z) < ∞, where dA is area measure on C normalized so that A(D) = 1. A closed subspace I of A is said to be invariant if zf(z) ∈ I for all f ∈ I. A sequence Z = {aj} of points in D is called an A zero set if there exists a non-zero function f ∈ A such that f vanishes on Z, counting multiplicities. Every A zero set generates an invariant subspace I Z consisting of all functions in A p that vanish on Z, counting multiplicities. Recently, Korenblum and Zhu [4] exhibited several classes of complemented invariant subspaces I in A. By definition, I is complemented in A if there exists a bounded projection Q from A onto I. In particular, Korenblum and Zhu considered the subspaces I Z for a special class of zero sequences Z. Recall that a sequence {aj} of distinct points in the disc is an A (H) interpolation sequence if for every sequence {wj} ⊂ C with ∑ (1− |aj |)|wj | < ∞ (∑ (1− |aj |)|wj | < ∞ ) there exists a function f ∈ A (f ∈ H) such that f(aj) = wj for all j. Recall that every H interpolation sequence is an A interpolation set (see, e.g., [5]); the converse implication does not hold. Further results about the interpolation sequences can be found in the monographs [3] and [2]. Theorem 1 (Korenblum and Zhu [4]). Suppose that 0 < p < ∞ and Z is an A interpolation sequence. Then the invariant subspace I Z is complemented in A . Received by the editors August 10, 2005 and, in revised form, August 22, 2005. 2000 Mathematics Subject Classification. Primary 47A15; Secondary 30D55, 30H05.