The Banach algebras $AC(σ)$ and $BV(σ)$

Abstract

The spaces BV (σ) and AC(σ) were introduced as part of a program to find a general theory which covers both wellbounded 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.

Approximation in $$AC(\sigma )$$

“…An important fact is the following result from [12] that says that being absolutely continuous is a local property. We shall say that a set U is a compact neighbourhood of a point x ∈ (with respect to ) if there exists an open neighbourhood V ⊆ ℂ of…”

“…We shall denote the Lipschitz functions on by Lip( ). This is a Banach algebra under the norm ‖f ‖ Lip( ) = ‖f ‖ ∞ + L (f ), where (For the degenerate case of a singleton set, we set L (f ) to be zero for all f.) In the following proposition [12,Theorem 3.11], C stands for the variation constant of , which is the variation of the identity function (z) = z over (considered as a subset of ℂ ). If lies inside a rectangle, then C is at most the sum of the width plus the height of the rectangle.…”

“…Unfortunately, depending on , there may be Lipschitz functions which are not absolutely continuous. Examples are given in [5,12].…”

“…2, we shall briefly give the main definitions and properties of the spaces involved. A more detailed account of the spaces BV( ) and AC( ) can be found in [12]. Note that the definitions given here involve a significant simplification of the original ones used in [5].…”