Joint spectrum for quasi-solvable Lie algebras of operators

TL;DR

This paper extends the concept of joint spectra from solvable to quasi-solvable Lie algebras of operators on Banach spaces, establishing key spectral properties and uniqueness of the extension.

Contribution

It introduces a method to extend the joint spectrum to quasi-solvable Lie algebras, preserving spectral properties and ensuring uniqueness.

Findings

Extended joint spectrum maintains spectral properties.

Extension is uniquely determined by the original spectrum.

Provides a framework for spectral analysis of quasi-solvable Lie algebras.

Abstract

Given a complex Banach space $X$ and a joint spectrum for complex solvable finite dimensional Lie algebras of operators defined on $X$, we extend this joint spectrum to quasi-solvable Lie algebras of operators, and we prove the main spectral properties of the extended joint spectrum. We also show that this construction is uniquely determined by the original joint spectrum.