A Finite Multiplicity Helson-Lowdenslager-De Branges Theorem

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This paper proves two theorems. The first of these simplifies and lends clarity to the previous characterizations of the invariant subspaces of S, the operator of multiplication by the coordinate function z, on L²(T;Cⁿ), where T is the unit circle, by characterizing the invariant subspaces of Sⁿ on scalar valued Lᵖ (0<p≤∞) thereby eliminating range functions and partial isometries. It also gives precise conditions as to when the operator shall be a pure shift and describes the precise nature of the wandering vectors and the doubly invariant subspaces. The second theorem describes the contractively contained Hilbert spaces in Lᵖ that are simply invariant under Sⁿ thereby generalizing the first theorem.