A Finite Multiplicity Helson-Lowdenslager-De Branges Theorem

TL;DR

This paper simplifies the understanding of invariant subspaces of multiplication operators on vector-valued $L^p$ spaces, providing clearer characterizations and conditions for pure shifts and invariant subspaces.

Contribution

It offers new, simplified characterizations of invariant subspaces of the multiplication operator on $L^p$ spaces, removing the need for range functions and partial isometries.

Findings

Characterizes invariant subspaces of $S^n$ on scalar $L^p$ spaces.

Provides conditions for when the operator is a pure shift.

Describes the structure of doubly invariant subspaces.

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.