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Towards spectral descriptions of cyclic functions

Miguel Monsalve, Daniel Seco

TL;DR

The paper develops a spectral framework to relate cyclicity of inner functions to the spectrum of the operator V_a = M^*_{a-z}M_{a-z} on weighted Hardy spaces H^2_ω. It derives a recurrence for eigenfunctions, characterizes the essential spectrum via Jacobi matrix perturbations, and completely describes the point spectrum and eigenfunctions for Bergman type spaces A^2_β, including explicit formulas and cases depending on a and α. The Dirichlet space case presents significant technical obstacles, with partial progress and open problems due to nonhomogeneous differential equations. The results illuminate how spectral data reflect cyclicity properties and motivate further connections to approximation theory, Toeplitz operators, and multiplier theory across RKHS.

Abstract

We build on a characterization of inner functions $f$ due to Le, in terms of the spectral properties of the operator $V=M_f^*M_f$ and study to what extent the cyclicity on weighted Hardy spaces $H^2_ω$ of the function $z \mapsto a-z$ can be inferred from the spectral properties of analogous operators $V_a$. We describe several properties of the spectra that hold in a large class of spaces and then, we focus on the particular case of Bergman-type spaces, for which we describe completely the spectrum of such operators and find all eigenfunctions.

Towards spectral descriptions of cyclic functions

TL;DR

The paper develops a spectral framework to relate cyclicity of inner functions to the spectrum of the operator V_a = M^*_{a-z}M_{a-z} on weighted Hardy spaces H^2_ω. It derives a recurrence for eigenfunctions, characterizes the essential spectrum via Jacobi matrix perturbations, and completely describes the point spectrum and eigenfunctions for Bergman type spaces A^2_β, including explicit formulas and cases depending on a and α. The Dirichlet space case presents significant technical obstacles, with partial progress and open problems due to nonhomogeneous differential equations. The results illuminate how spectral data reflect cyclicity properties and motivate further connections to approximation theory, Toeplitz operators, and multiplier theory across RKHS.

Abstract

We build on a characterization of inner functions due to Le, in terms of the spectral properties of the operator and study to what extent the cyclicity on weighted Hardy spaces of the function can be inferred from the spectral properties of analogous operators . We describe several properties of the spectra that hold in a large class of spaces and then, we focus on the particular case of Bergman-type spaces, for which we describe completely the spectrum of such operators and find all eigenfunctions.
Paper Structure (7 sections, 10 theorems, 90 equations)

This paper contains 7 sections, 10 theorems, 90 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Recurrence relations
  4. Essential and point spectrum
  5. A complete description of point spectrum and eigenfunctions
  6. The Dirichlet space case
  7. Further remarks

Key Result

Theorem 1.1

Let $\omega$ be valid and $f$ be a multiplier on $H^2_\omega$. Then the following are equivalent:

Theorems & Definitions (21)

  • Theorem 1.1: Le
  • Theorem 1.2
  • Lemma 3.1
  • proof
  • Lemma 4.1
  • proof
  • Corollary 4.2
  • proof
  • Theorem 4.3
  • Theorem 4.4
  • ...and 11 more