Blaschke cocycles and generators

Abstract

Using a local product decomposition, we establish a certain class of Blaschke cocycles with the property that a simply invariant subspace has a single generator if and only if its cocycle is cohomologous to one of this class. Some applications are also obtained. We show, among other things, every simply invariant subspace is approximated by a singly generated one as near as desired. 1. Preliminaries. Let Γ be a dense subgroup of the real line R, endowed with the discrete topology, and let K be the dual group of Γ. For each t in /?, et denotes the element of K defined by et(λ) = eiλt for any λ in Γ. Then the mapping from t to et embeds R continuously onto a dense subgroup of K. Choose and fix a positive γ in Γ, and let Kγ be the subgroup consisting of all x in K such that x(γ) = 1. Then K may be identified measure theoretically, and almost topologically, with Kγ x [0,2π/γ) via the mapping y + es to (y9s). We assume, for simplicity, that 2π lies in Γ throughout the paper. Thus K may be regarded as K2π x [0,1). This local product decomposition is very useful for understanding the group K. We denote by σ and θ the normalized Haar measures on K and K2π, respectively. Then is carried by the above mapping to the restriction of dσ x dt to K2π x