Invariant subspaces for finite index shifts in Hardy spaces

TL;DR

This paper characterizes shift-invariant subspaces in finite direct sums of Hardy spaces, extending Beurling's theorem, and shows that finite defect contractions have nontrivial invariant subspaces.

Contribution

It provides a concrete Beurling-type theorem for finite index shifts in Hardy space sums using determinantal operators.

Findings

Characterization of shift-invariant subspaces in finite sums of Hardy spaces.

Every invariant subspace of codimension at least two is contained in a larger invariant subspace.

Finite defect contractions have nontrivial invariant subspaces.

Abstract

Let $\mathbb H$ be the finite direct sums of $H^2(\mathbb D)$. In this paper, we give a characterization of the closed subspaces of $\mathbb H$ which are invariant under the shift, thus obtaining a concrete Beurling-type theorem for the finite index shift. This characterization presents any such a subspace as the finite intersection, up to an inner function, of pre-images of a closed shift-invariant subspace of $H^2(\mathbb D)$ under ``determinantal operators'' from $\mathbb H$ to $H^2(\mathbb D)$, that is, continuous linear operators which intertwine the shifts and appear as determinants of matrices with entries given by bounded holomorphic functions. With simple algebraic manipulations we provide a direct proof that every invariant closed subspace of codimension at least two sits into a non-trivial closed invariant subspace. As a consequence every contraction with finite defect has a nontrivial closed invariant subspace.