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Invariant subspaces for finite index shifts in Hardy spaces

Filippo Bracci, Eva A. Gallardo-Gutiérrez

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.

Invariant subspaces for finite index shifts in Hardy spaces

Abstract

Let be the finite direct sums of . In this paper, we give a characterization of the closed subspaces of 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 under ``determinantal operators'' from to , 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.

Paper Structure

This paper contains 8 sections, 14 theorems, 107 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Closed $S$-invariant spaces associated to matrices
  4. Case $\det A\not\equiv 0$
  5. Case $\det A\equiv 0$
  6. The spaces $\mathcal{N}_A$ and $\mathcal{N}_{J,A}$ are not maximal for the shift
  7. The Beurling-Lax matrix of an invariant subspace
  8. The structure theorem

Key Result

Theorem 1.1

Let $N\neq\{0\}$ be a closed subspace of $\mathbb H$. Then $N$ is $S$-invariant if and only if there exist inner functions $\varphi, \phi$ and determinantal subspaces $\mathcal{Q}_\varphi$ and $\mathcal{Q}_0$ such that either or

Theorems & Definitions (41)

  • Theorem 1.1
  • Theorem 1.2
  • Corollary 1.3
  • Lemma 2.1
  • Definition 2.2
  • Lemma 2.3
  • proof
  • Definition 3.1
  • Definition 3.2
  • Lemma 3.3
  • ...and 31 more