Closed operator functional calculus in Banach modules and applications

TL;DR

This paper develops a closed operator functional calculus for Banach modules over the group algebra and demonstrates its applications, including spectral mapping theorems and spectrum estimates for perturbed differential operators.

Contribution

It introduces a new closed operator functional calculus in Banach modules over $L^1(\mathbb R)$ and applies it to spectral analysis and perturbation estimates.

Findings

Spectral mapping theorem for operators in the calculus

Estimate for spectrum of perturbed differential operators

Application to spectral analysis in Banach modules

Abstract

We describe a closed operator functional calculus in Banach modules over the group algebra $L^1(\mathbb R)$ and illustrate its usefulness with a few applications. In particular, we deduce a spectral mapping theorem for operators in the functional calculus, which generalizes some of the known results. We also obtain an estimate for the spectrum of a perturbed differential operator in a certain class.