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On the essential spectra of Toeplitz operators on Bergman spaces of the bi-disc

Uǧur Gül

Abstract

In this note we deal with the problem of determining the essential spectrum of a Toeplitz operator $T_{f}:A^{2}(\mathbb{D}^{2})\rightarrow A^{2}(\mathbb{D}^{2})$ acting on the Bergman space $A^{2}(\mathbb{D}^{2})$ of the bi-disc whose symbol $f\in C(\overline{\mathbb{D}^{2}})$ lies in the space of continuous functions on the closure $\overline{\mathbb{D}^{2}}$ of the bi-disc.

On the essential spectra of Toeplitz operators on Bergman spaces of the bi-disc

Abstract

In this note we deal with the problem of determining the essential spectrum of a Toeplitz operator acting on the Bergman space of the bi-disc whose symbol lies in the space of continuous functions on the closure of the bi-disc.
Paper Structure (3 sections, 5 theorems, 24 equations)

This paper contains 3 sections, 5 theorems, 24 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Main Results

Key Result

Theorem 2.1

Murphy A bounded linear operator $T$ on a Hilbert space $H$ is Fredholm if and only if $T+K(H)$ is invertible in the quotient algebra $B(H)/K(H)$, where $K(H)$ is the algebra of all compact operators on $H$.

Theorems & Definitions (7)

  • Theorem 2.1
  • Theorem 2.2
  • Proposition 2.3
  • Proposition 3.1
  • proof
  • Theorem 3.2
  • proof