A spectral theory for solvable Lie algebras of operators

TL;DR

This paper introduces a new spectral theory for complex solvable Lie algebras of operators on Banach spaces, extending the Taylor joint spectrum to non-commuting cases.

Contribution

It develops a joint spectrum concept for solvable Lie algebras of operators, generalizing existing spectral notions for commuting operators.

Findings

Defines a joint spectrum for solvable Lie algebras of operators.

Extends the Taylor joint spectrum to non-commuting operator algebras.

Provides a framework for spectral analysis of solvable Lie algebra representations.

Abstract

The main objective of this paper is to develop a notion of joint spectrum for complex solvable Lie algebras of operators acting on a Banach space, which generalizes the Taylor joint spectrum (T.J.S.) for several commuting operators.