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Cyclicity of the shift operator and a related completeness problem in de Branges-Rovnyak spaces

Emmanuel Fricain, Romain Lebreton

Abstract

In this paper, we study the cyclic vectors of the shift operator $S_b$ acting on de Branges-Rovnyak space $\mathcal H(b)$ associated to a non-extreme point of the closed unit ball of $H^\infty$. We highlight an interesting link with a completeness problem that we study using the Cauchy transform. This enables us to obtain some nice consequences on cyclicity.

Cyclicity of the shift operator and a related completeness problem in de Branges-Rovnyak spaces

Abstract

In this paper, we study the cyclic vectors of the shift operator acting on de Branges-Rovnyak space associated to a non-extreme point of the closed unit ball of . We highlight an interesting link with a completeness problem that we study using the Cauchy transform. This enables us to obtain some nice consequences on cyclicity.
Paper Structure (9 sections, 20 theorems, 91 equations)

This paper contains 9 sections, 20 theorems, 91 equations.

Table of Contents

  1. Introduction
  2. Preliminaries and useful analytic tools
  3. De Branges-Rovnyak spaces Hb
  4. Cauchy transform and distribution function
  5. A completeness problem related to cyclicity
  6. A completeness problem in the Hardy space
  7. A completeness problem in
  8. Main result on the completeness problem
  9. Some consequences on cyclicity

Key Result

Lemma 2.1

Let $\alpha\in\mathbb{T}$. Then

Theorems & Definitions (45)

  • Lemma 2.1
  • proof
  • Theorem 2.2
  • proof
  • Lemma 2.3
  • proof
  • Lemma 2.4
  • proof
  • Theorem 2.5: Kolmogorov
  • proof
  • ...and 35 more