Simply Invariant Subspaces and Generalized Analytic Functions
T. P. Srinivasan · Proceedings of the American Mathematical Society · 1965
Let (X, m) be a probability measure space and A a subalgebra of Lxidm) on which m is "multiplicative," meaning, I fgdm= I f dm j g dm 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.