A Finite Multiplicity Helson-Lowdenslager-De Branges Theorem

Abstract

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 $L^2(\mathbb{T};\mathbb{C}^n)$, where $\mathbb{T}$ is the unit circle, by characterizing the invariant subspaces of $S^n$ on scalar valued $L^p$ ($0<p\le\infty$) 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 $L^p$ that are simply invariant under $S^n$ thereby generalizing the first theorem.