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Hankel matrices acting on the Dirichlet space

Guanlong Bao, Kunyu Guo, Fangmei Sun, Zipeng Wang

Abstract

The characterization of the boundedness of operators induced by Hankel matrices on analytic function spaces can be traced back to the work of Z. Nehari and H. Widom on the Hardy space, and has been extensively studied on many other analytic function spaces recently. However, this question remains open in the context of the Dirichlet space [20]. By Carleson measures, the Widom type condition and the reproducing kernel thesis, this paper provides a comprehensive solution to this question. As a beneficial product, characterizations of the boundedness and compactness of operators induced by Cesàro type matrices on the Dirichlet space are given. In addition, we also show that a random Dirichlet function almost surely induces a compact Hankel type operator on the Dirichlet space.

Hankel matrices acting on the Dirichlet space

Abstract

The characterization of the boundedness of operators induced by Hankel matrices on analytic function spaces can be traced back to the work of Z. Nehari and H. Widom on the Hardy space, and has been extensively studied on many other analytic function spaces recently. However, this question remains open in the context of the Dirichlet space [20]. By Carleson measures, the Widom type condition and the reproducing kernel thesis, this paper provides a comprehensive solution to this question. As a beneficial product, characterizations of the boundedness and compactness of operators induced by Cesàro type matrices on the Dirichlet space are given. In addition, we also show that a random Dirichlet function almost surely induces a compact Hankel type operator on the Dirichlet space.
Paper Structure (6 sections, 15 theorems, 96 equations)

This paper contains 6 sections, 15 theorems, 96 equations.

Table of Contents

  1. Introduction
  2. Bounded Hankel type operators $\mathcal{H}_{\boldsymbol{\lambda}}$ on $\mathcal{D}$
  3. Bounded Hankel type operators $\mathcal{H}_\mu$ and Cesàro type operators on the Dirichlet space
  4. Compact Hankel and Cesàro type operators on $\mathcal{D}$
  5. Random Hankel type operators on $\mathcal{D}$ and a result related to Rudin's $\Lambda(p)$ set
  6. Final remarks

Key Result

Theorem 1.1

Suppose $\boldsymbol{\lambda}=\{\lambda_n\}_{n\in\mathbb{N}}$ is a sequence of complex numbers. Then the Hankel type operator $\mathcal{H}_{\bm\lambda}$ is bounded on the Dirichlet space $\mathcal{D}$ if and only if $h_{\overline{\bm \lambda}}$ is analytic on $\mathbb{D}$ and the measure $|h_{\overl

Theorems & Definitions (27)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Corollary 1.5
  • Theorem 1
  • proof : Proof of Theorem \ref{['main00']}
  • Proposition 2.1
  • proof
  • Theorem 2
  • ...and 17 more