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 $\rho \text{\hspace{0.17em}⁣}:L\to L(X)$, 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.