Spectrum of differential operators with elliptic adjoint on a scale of localized Sobolev spaces

TL;DR

This paper investigates the spectrum of constant coefficient differential operators on localized Sobolev spaces, revealing unique spectral properties in Fréchet spaces that differ from classical Banach space results.

Contribution

It provides a comprehensive analysis of the spectrum on localized Sobolev spaces, highlighting the absence of resolvent points and proposing a new spectral definition for Fréchet spaces.

Findings

No complex number in the resolvent set of such operators

Distinct behavior of spectrum types in Fréchet spaces

Potential new framework for spectral theory in Fréchet spaces

Abstract

In this paper we provide a complete study of the spectrum of a constant coefficients differential operator on a scale of localized Sobolev spaces, $H^{s}_{loc}(I),$ which are Fr\'echet spaces. This is quite different from what we find in the literature, where all the relevant results are concerned with spectrum on Banach spaces. Our aim is to understand the behavior of all the three types of spectrum (point, residual and continuous) and the relation between them and those of the dual operator. The main result we present shows that there is no complex number in the resolvent set of such operators, which suggest a new way to define spectrum if we want to reproduce the classical theorems of the Spectral Theory in Fr\'echet spaces.