Smooth Functional Calculus and Spectral Theorem in Banach Spaces
Functional Analysis
2025-04-01 v2 Mathematical Physics
math.MP
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.
Cite
@article{arxiv.2410.20725,
title = {Smooth Functional Calculus and Spectral Theorem in Banach Spaces},
author = {Luis A. Cedeño-Pérez and Hernando Quevedo},
journal= {arXiv preprint arXiv:2410.20725},
year = {2025}
}
Comments
Typos corrected, a proof changed, comments added