Simply Invariant Subspaces and Generalized Analytic Functions1

for all/, g E A. Let A 0 be the set of functions in A with// dm = 0. Denote by Hpidm) the closure [^4]p of ^4 in Lpidm), p = i, 2 and by Haidm) the weak* closure |yl]* of A in P°°(¿wi). We shall drop the parenthesis (dm), in the future, while referring to Lpidm) Hpidm), etc. The functions in Hp we call generalized analytic functions. Say that a closed subspace ÜDÍ of Lp is simply invariant if [^o.Sft]? CSD? ahd 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 qEL" and | q\ = 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 P^theorem follows from the P2-theorem and the following two conditions are sufficient for the truth of the P2-theorem: Hi. A -\-A is dense in P2 where the bar denotes complex conjugation. H2. If fEL\f£0 and if ffg dm = 0 for all gEA0 then/=c a.e. for some constant c. It turns out that these two conditions are necessary as well for the validity of the P2-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 [S] to derive the P ^invariant subspace theorem from the P2-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 P1invariant subspace theorem in [5] had some gaps. We rederive this theorem here (Corollary 2.5) for completeness. The P^theorem in