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Hilbert-Schmidt Hankel operators with harmonic symbols on the Bergman space of strongly pseudoconvex domains in $\mathbb{C}^n$

Timothy G. Clos

Abstract

We characterize Hilbert-Schmidt Hankel operators on the Bergman spaces of smooth bounded strongly pseudoconvex domains in $\mathbb{C}^n$ for $n \geq 2$. We consider harmonic symbols of class $C^3$ up to the closure of the domain and show $H_φ$ is Hilbert-Schmidt if and only if $φ$ is holomorphic on the domain.

Hilbert-Schmidt Hankel operators with harmonic symbols on the Bergman space of strongly pseudoconvex domains in $\mathbb{C}^n$

Abstract

We characterize Hilbert-Schmidt Hankel operators on the Bergman spaces of smooth bounded strongly pseudoconvex domains in for . We consider harmonic symbols of class up to the closure of the domain and show is Hilbert-Schmidt if and only if is holomorphic on the domain.

Paper Structure

This paper contains 5 sections, 4 theorems, 35 equations.

Table of Contents

  1. Introduction
  2. The Main Result
  3. Proof of Theorem \ref{['prelim']}
  4. Proof of Corollary \ref{['cor1']}
  5. Acknowlegments

Key Result

Theorem 1

Let $\Omega\subset \mathbb{C}^n$ be a $C^{\infty}$-smooth bounded strongly pseuodoconvex domain and $n\geq 2$. Suppose $\phi\in C^3(\overline{\Omega})$ and $H_{\phi}$ is Hilbert-Schmidt on $A^2(\Omega)$. Then $\overline{\partial}_b(\phi)=0$ on $b\Omega$. That is, the tangential component of $\overli

Theorems & Definitions (10)

  • Theorem 1
  • Corollary 1
  • Proposition 1
  • proof
  • Lemma 1
  • proof
  • proof
  • proof
  • Remark 1
  • Remark 2