A General Beurling-Helson-Lowdenslager Theorem on the Disk

TL;DR

This paper generalizes the classical Beurling-Helson-Lowdenslager theorem to a broad class of function spaces defined by new norms, extending the characterization of shift-invariant subspaces beyond traditional Hardy and Lebesgue spaces.

Contribution

It introduces a new, simple proof extending the invariant subspace theorem to spaces $H^{ ext{alpha}}$ and $L^{ ext{alpha}}$ defined by general norms.

Findings

Extended the invariant subspace theorem to new norm-defined spaces

Provided a simple, self-contained proof approach

Generalized classical results to broader function spaces

Abstract

The classical Beurling-Helson-Lowdenslager theorem characterizes the shift-invariant subspaces of the Hardy space $H^{2}$ and of the Lebesgue space $L^{2}$. In this paper, which is self-contained, we define a very general class of norms $\alpha$ and define spaces $H^{\alpha}$ and $L^{\alpha}.$ 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 $\left \Vert {}\right \Vert _{p}$-results for our more general norms.