JOINT SPECTRA OF REPRESENTATIONS OF LIE ALGEBRAS BY COMPACT OPERATORS

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 Słodkowski, the Fredholm, the split and the Fredholm split joint spectra of the representation $\rho$ are computed.