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Smooth Functional Calculus and Spectral Theorem in Banach Spaces

Luis A. Cedeño-Pérez, Hernando Quevedo

TL;DR

The article generalizes the spectral representation to Banach spaces by introducing projection families and a two-stage functional calculus. It develops a Smooth Functional Calculus via the Cauchy-Pompeiu framework and extends it to a Continuous Calculus, enabling a spectral decomposition in Banach algebras and for operators on Banach spaces through associated vector measures μ^x and μ^T. The main contributions are the existence and uniqueness of these spectral families, the extension to L^1(μ) and L^∞(μ) integrals, and the unification of the Hilbert-space spectral theorem within a broad Banach-space context. This work provides a robust, measure-theoretic approach to spectral representation beyond Hilbert spaces, with potential applications in analysis and quantum information contexts.

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.

Smooth Functional Calculus and Spectral Theorem in Banach Spaces

TL;DR

The article generalizes the spectral representation to Banach spaces by introducing projection families and a two-stage functional calculus. It develops a Smooth Functional Calculus via the Cauchy-Pompeiu framework and extends it to a Continuous Calculus, enabling a spectral decomposition in Banach algebras and for operators on Banach spaces through associated vector measures μ^x and μ^T. The main contributions are the existence and uniqueness of these spectral families, the extension to L^1(μ) and L^∞(μ) integrals, and the unification of the Hilbert-space spectral theorem within a broad Banach-space context. This work provides a robust, measure-theoretic approach to spectral representation beyond Hilbert spaces, with potential applications in analysis and quantum information contexts.

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.

Paper Structure

This paper contains 14 sections, 27 theorems, 103 equations, 1 table.

Table of Contents

  1. Introduction
  2. Geometric Integration
  3. Cauchy-Pompeiu Formula
  4. Coarea Formula
  5. Projection Families and Integration
  6. Vector Integration
  7. Projection Families
  8. Smooth and Continuous Functional Calculi
  9. Vector Cauchy-Pompeiu Formula
  10. Regular Elements and Smooth Functional Calculus
  11. Continuous Functional Calculus
  12. Spectral Theorem in Banach Spaces
  13. Spectral Theorem in Banach Algebras
  14. Spectral Theorem for Operators in Banach Spaces

Key Result

Theorem 2.1

Let $\Omega\subset\mathbb{C}$ be an open set with compact closure, $u\in C^{\infty}(\Omega)$ and $\omega\subset\Omega$ an open set with smooth boundary such that $\partial\omega\subset\Omega$. The formula is valid for any $\lambda\in\omega$.

Theorems & Definitions (57)

  • Theorem 2.1: Cauchy-Pompeiu Formula
  • proof
  • Corollary 2.1.1
  • proof
  • Theorem 2.2: Cauchy-Pompeiu
  • proof
  • Theorem 2.3: Coarea Formula
  • proof
  • Corollary 2.3.1
  • proof
  • ...and 47 more