A general Beurling–Helson–Lowdenslager theorem on the disk

Abstract

The classical Beurling–Helson–Lowdenslager theorem characterizes the shift-invariant subspaces of the Hardy space H2 and of the Lebesgue space L2. In this paper, which is self-contained, we define a very general class of norms α and define spaces Hα and Lα. We then extend the Beurling–Helson–Lowdenslager invariant subspace theorem. The idea of the proof is new and quite simple; most of the details involve extending basic well-known ‖⋅‖p-results for our more general norms.