The Banach algebras $AC(\sigma)$ and $BV(\sigma)$

TL;DR

This paper simplifies the definitions and explores the main properties of the Banach algebras $AC(\sigma)$ and $BV(\sigma)$, which unify theories of well-bounded and trigonometrically well-bounded operators on Banach spaces.

Contribution

It provides a simplified, self-contained exposition of the properties of $AC(\sigma)$ and $BV(\sigma)$ spaces, advancing the understanding of their role in operator theory.

Findings

Simplified definitions of $AC(\sigma)$ and $BV(\sigma)$ spaces.

Clarification of their main properties.

Foundation for further research in operator theory.

Abstract

The spaces $BV(\sigma)$ and $AC(\sigma)$ were introduced as part of a program to find a general theory which covers both well-bounded operators and trigonometrically well-bounded operators acting on a Banach space. Since their initial appearance it has become clear that the definitions could be simplified somewhat. In this paper we give a self-contained exposition of the main properties of these spaces using this simplified approach.