Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Spectrum of Weighted Composition Operators. Part IX. The spectrum and essential spectra of some weighted composition operators on uniform algebras

Arkady Kitover, Mehmet Orhon

Abstract

We obtain some results about the spectrum and the upper semi-Fredholm spectrum of weighted composition operators on uniform algebras, assuming that the corresponding map maps the Shilov boundary onto itself. In particular, it follows from our results that in the case of analytic uniform algebras the spectrum is a connected rotation invariant subset of the complex plane, and that the upper semi-Fredholm spectrum is rotation invariant as well.

Spectrum of Weighted Composition Operators. Part IX. The spectrum and essential spectra of some weighted composition operators on uniform algebras

Abstract

We obtain some results about the spectrum and the upper semi-Fredholm spectrum of weighted composition operators on uniform algebras, assuming that the corresponding map maps the Shilov boundary onto itself. In particular, it follows from our results that in the case of analytic uniform algebras the spectrum is a connected rotation invariant subset of the complex plane, and that the upper semi-Fredholm spectrum is rotation invariant as well.
Paper Structure (4 sections, 16 theorems, 37 equations)

This paper contains 4 sections, 16 theorems, 37 equations.

Table of Contents

  1. Introduction and preliminaries
  2. The main results
  3. Essential spectra of weighted automorphisms of disc-algebra
  4. Examples

Key Result

Lemma 1.1

Let $A$ be a unital uniform algebra and $t \in \partial A$ is a $p$-point. Then the characteristic function $\chi_{\{t\}}$ of the singleton $\{t\}$ can be identified with an element of the second dual $A^{\prime \prime}$.

Theorems & Definitions (35)

  • Lemma 1.1
  • proof
  • Theorem 2.1
  • proof
  • Remark 2.2
  • Corollary 2.3
  • proof
  • Definition 2.4
  • Corollary 2.5
  • Theorem 2.6
  • ...and 25 more