Joint spectra of representations of Lie algebras by compact operators

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

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