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Counting functions for Dirichlet series and compactness of composition operators

Frédéric Bayart

Abstract

We give a sufficient condition for a composition operator with positive characteristic to be compact on the Hardy space of Dirichlet series.

Counting functions for Dirichlet series and compactness of composition operators

Abstract

We give a sufficient condition for a composition operator with positive characteristic to be compact on the Hardy space of Dirichlet series.
Paper Structure (8 sections, 5 theorems, 33 equations)

This paper contains 8 sections, 5 theorems, 33 equations.

Table of Contents

  1. Introduction
  2. Description of the results
  3. Background material
  4. On counting functions for Dirichlet series
  5. Compact composition operators
  6. Compactness on Hardy spaces
  7. Compactness on Bergman spaces
  8. Concluding remarks

Key Result

Theorem 1.1

Let $\varphi\in{\mathcal{G}}_{\geq 1}$ and let us assume that $\mathcal{N}_\varphi(w)=o(\Re e(w))$ as $\Re e(w)$ tends to $0$. Then $C_\varphi$ is compact on $\mathcal{H}^2$.

Theorems & Definitions (14)

  • Theorem 1.1
  • Proposition 2.1
  • proof
  • Corollary 2.2
  • proof
  • proof : Proof of Theorem \ref{['thm:main']}
  • Remark 3.1
  • Remark 3.2
  • Theorem 3.3
  • proof
  • ...and 4 more