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On spectra of Hankel operators on the polydisc

Zeljko Cuckovic, Zhenghui Huo, Sonmez Sahutoglu

TL;DR

The paper addresses spectral properties of Hankel operators on the Bergman space of the polydisc, focusing on when the essential spectrum of the Hermitian square of a Hankel operator contains intervals and on explicit spectra for monomial symbols. The authors develop two sufficient conditions yielding interval support in the essential spectrum: a boundary-driven separable-symbol analysis and a general non-constant boundary norm condition, and they derive a complete eigen-spectrum for monomials. The methods combine Weyl's criterion, product-domain decompositions, and detailed estimates of cross-terms to connect boundary behavior to spectral values. The results advance understanding of Hankel–Toeplitz semicommutators in several variables and could inform boundary regularity questions in several complex variables.

Abstract

We give sufficient conditions for the essential spectrum of the Hermitian square of a class of Hankel operators on the Bergman space of the polydisc to contain intervals. We also compute the spectrum in case the symbol is a monomial.

On spectra of Hankel operators on the polydisc

TL;DR

The paper addresses spectral properties of Hankel operators on the Bergman space of the polydisc, focusing on when the essential spectrum of the Hermitian square of a Hankel operator contains intervals and on explicit spectra for monomial symbols. The authors develop two sufficient conditions yielding interval support in the essential spectrum: a boundary-driven separable-symbol analysis and a general non-constant boundary norm condition, and they derive a complete eigen-spectrum for monomials. The methods combine Weyl's criterion, product-domain decompositions, and detailed estimates of cross-terms to connect boundary behavior to spectral values. The results advance understanding of Hankel–Toeplitz semicommutators in several variables and could inform boundary regularity questions in several complex variables.

Abstract

We give sufficient conditions for the essential spectrum of the Hermitian square of a class of Hankel operators on the Bergman space of the polydisc to contain intervals. We also compute the spectrum in case the symbol is a monomial.
Paper Structure (3 sections, 13 theorems, 84 equations)

This paper contains 3 sections, 13 theorems, 84 equations.

Table of Contents

  1. Main Results
  2. Proof of Theorems \ref{['ThmProdSym']} and \ref{['ThmGenSym']}
  3. Proof of Theorem \ref{['ThmMonSym']}

Key Result

Theorem 1

Let $\varphi\in C(\overline{\mathbb{D}^{n-1}}),\chi\in C(\overline{\mathbb{D}})$ and $\psi(z',z_n)=\varphi(z')\chi(z_n)$ for $z'\in \overline{\mathbb{D}^{n-1}},$ and $z_n\in \overline{\mathbb{D}}$. Then the essential spectrum of $H^*_{\psi}H_{\psi}$ contains the set

Theorems & Definitions (27)

  • Theorem 1
  • Corollary 1
  • Theorem 2
  • Corollary 2
  • Example 1
  • Theorem 3
  • Corollary 3
  • Corollary 4
  • Corollary 5
  • Theorem A: Weyl's Criterion
  • ...and 17 more