Characterization of certain invariant subspaces of H^p and L^p spaces derived from logmodular algebras

Samuel Merrill and Nand Lal

Vol. 30 (1969), No. 2, 463–474

DOI: 10.2140/pjm.1969.30.463

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 ≦∞, let H^p(dm) denote the closure of A in L^p(dm) ( w^∗ closure for p = ∞). A closed subspace M of H^p(dm) or L^p(dm) is called invariant if f ∈ M and g ∈ A imply that fg ∈ 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.

Mathematical Subject Classification

Primary: 46.55

Milestones

Received: 4 October 1968

Published: 1 August 1969