Spectrum of weighted composition operators. Part XII. Kamowitz - Scheinberg theorem revisited

TL;DR

This paper investigates the spectrum of weighted automorphisms $T=wU$ on unital, commutative, semisimple Banach algebras by analyzing the associated weighted composition operator $\breve{T}$ on the Shilov boundary. A key result provides a summability condition $\sum_{n=-\infty}^{\infty} \frac{|\ln\| (\breve{T})^{n}\| |}{1+n^{2}}<\infty$ that ensures $\mathds{T} \subseteq \sigma(T)$ when $U^n\neq I$. Under additional boundary hypotheses (no $\varphi$-periodic points and every boundary point is a peak point for $\check{A}$), the unit circle is contained in the spectrum, and when $\sigma(\breve{T})\subseteq \mathds{T}$ with invertible weights the spectrum is connected. The work also develops special cases where $\sigma(T)=\sigma(\breve{T})$, and provides extensive examples (including Lip$_α$ domains and the Wiener algebra) to illustrate when $\sigma(T)$ is an annulus or circle and how it relates to invariant measures and weight functions.

Abstract

The well-known Kamowitz - Scheinberg theorem states that if $U$ is an automorphism of a commutative semi-simple Banach algebra and $U^n \neq I, n \in \mathds{N}$, then the spectrum of $U$ contains the unit circle. In this paper we present some results about the spectrum of weighted automorphisms of unital commutative semi-simple Banach algebras that considerably strengthen the statement of the Kamowitz - Scheinberg theorem.

Theorems & Definitions (29)

• Theorem 1.1
• Theorem 2.1
• proof
• Corollary 2.2
• proof
• Theorem 2.4
• proof
• Theorem 2.5
• proof
• Corollary 2.6