Smooth Functional Calculus and Spectral Theorem in Banach Spaces

TL;DR

This paper extends the spectral theorem to Banach algebras and operators between Banach spaces by developing a smooth and continuous functional calculus using projection families.

Contribution

It introduces a generalized spectral theorem for Banach algebras based on projection families and develops a smooth functional calculus via the Cauchy-Pompeiu formula.

Findings

Spectral theorem extended to Banach algebras

Developed smooth and continuous functional calculus

Generalized classical results for Hilbert space operators

Abstract

The notion of projection families generalizes the classical notions of vector- and operator-valued measures. We show that projection families are general enough to extend the Spectral Theorem to Banach algebras and operators between Banach spaces. To this end, we first develop a Smooth Functional Calculus in Banach algebras using the Cauchy-Pompeiu formula, which is further extended to a Continuous Functional Calculus. We also show that these theorems are proper generalizations of the usual result for operators between Hilbert spaces.