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 $H^2$ 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 ${\Vert \Vert }_p$-results for our more general norms.