Smooth Functional Calculus and Spectral Theorem in Banach Spaces
Luis Antonio Cedeño-Pérez, Hernando Quevedo · arXiv (Cornell University) · 2024
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.