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A General Beurling-Helson-Lowdenslager Theorem on the Disk

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 $α$ and define spaces $H^α$ and $L^α.$ 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.