Simply invariant subspaces and generalized analytic functions

Abstract

for allf, gGA. Let Ao be the set of functions in A with ff dm=0. Denote by HP(dm) the closure [A], of A in LP(dm), p=1, 2 and by H??(dm) the weak* closure [A ]* of A in L??(dm). We shall drop the parenthesis (dm), in the future, while referring to LP(dm) HP(dm), etc. The functions in HP we call generalized analytic functions. Say that a closed subspace P of LP is simply invariant if [AJ,D]p C 9? a:nd the inclusion is strict. For logmodular algebras A it was shown in [5] that the simply invariant subspaces of LP have the form qHP where qELP and I ql = 1 a.e. We shall refer to this result as the LPinvariant subspace theorem. The proof in [5] also shows that the logmodularity of A is inessential for the truth of this theorem, that the LI-theorem follows from the L2-theorem and the following two conditions are sufficient for the truth of the L2-theorem: H1. A +A is dense in L2 where the bar denotes complex conjugation. H2. If fCL',f >0 and if ffg dm=0 for all gEAo then f=c a.e. for some constant c. It turns out that these two conditions are necessary as well for the validity of the L2-theorem. One of our purposes in this paper is to prove this (Corollaries 1.1, 2.4). The key to our proof is a factorization theorem (Theorem 2) used (but not explicitly stated) by us in [5] to derive the L'-invariant subspace theorem from the L2-theorem. We derive on the way, from this factorization theorem, several consequences on generalized analytic functions which were proved by Hoffman [3] in the special case of logmodular algebras; Hoffman's machinery was different and more elaborate. Our proof of the Llinvariant subspace theorem in [5] had some gaps. We rederive this theorem here (Corollary 2.5) for completeness. The LI-theorem in