Characterization of certain invariant subspaces ofHpandLpspaces derived from logmodular algebras

Abstract

Let A = A(X) be a logmodular algebra and m a representing measure on X associated with a nontrivial Gleason part. For 1 ^ p ^ oo, let Hp(dm) denote the closure of A in Lp(dm) (w* closure for p — oo). A closed subspace M of Hp(dm) or Lp(dm) is called invariant if fe M and g e A imply that fg e M. The main result of this paper is a characterization of the invariant subspaces which satisfy a weaker hypothesis than that required in the usual form of the generalized Beurling theorem, as given by Hoffman or Srinivasan. For 1 ^ p ^ oo, let Ip be the subspace of functions in Hp(dm) vanishing on the Gleason part of m and let Am = ifeA: fdm — ok THEOREM. Let M be a closed invariant subspace of L 2(dm) such that the linear span of A mM is dense in M but the subspace R = {feM:fA_I°°M} is nontrivial and has the same support set E as M. Then M has the form χE F'(P)L for some unimodular function F. A modified form of the result holds for 1 <^ p <Ξ oo. This theorem is applied to give a complete characterization of the invariant subspaces of Lp(dm) when A is the standard algebra on the torus associated with a lexicographic ordering of the dual group and m is normalized Haar measure.