Joint spectra of representations of Lie algebras by compact operators

TL;DR

This paper computes various joint spectra for representations of finite-dimensional nilpotent Lie algebras on complex Banach spaces, where all operators are compact.

Contribution

It provides explicit calculations of Taylor, Slodkowski, Fredholm, split, and Fredholm split joint spectra for such Lie algebra representations.

Findings

Explicit formulas for joint spectra of nilpotent Lie algebra representations

Extension of spectral theory to compact operator representations

Unified approach to multiple joint spectra types

Abstract

Given $X$ a complex Banach space, $L$ a complex nilpotent finite dimensional Lie algebra, and $\rho\colon L\to L(X)$, a representation of $L$ in $X$ such that $\rho (l)\in K(X)$ for all $l\in L$, the Taylor, the Slodkowski, the Fredholm, the split and the Fredholm split joint spectra of the representation $\rho$ are computed.