Extensions of an $AC(\sigma)$ functional calculus

TL;DR

This paper investigates the extension of absolutely continuous functional calculi for operators on Banach spaces, showing it is generally possible on reflexive spaces but not on most nonreflexive spaces, with conditions for extension on interpolation spaces.

Contribution

It demonstrates the limitations of extending $ ext{AC}(\sigma)$ functional calculus on nonreflexive spaces and provides conditions for such extensions on interpolation spaces.

Findings

Extension is always possible on reflexive Banach spaces.

Extension generally impossible on most classical nonreflexive spaces.

Conditions are identified for extension on $L^p$ and similar spaces.

Abstract

On a reflexive Banach space $X$, if an operator $T$ admits a functional calculus for the absolutely continuous functions on its spectrum $\sigma(T) \subseteq \mathbb{R}$, then this functional calculus can always be extended to include all the functions of bounded variation. This need no longer be true on nonreflexive spaces. In this paper, it is shown that on most classical separable nonreflexive spaces, one can construct an example where such an extension is impossible. Sufficient conditions are also given which ensure that an extension of an $\AC$ functional calculus is possible for operators acting on families of interpolation spaces such as the $L^p$ spaces.