Dual properties and joint spectra for solvable Lie algebras of operators

Given $L$ a solvable Lie Algebra of operators acting on a Banach space $E$, we study the action of the opposite algebra of $L$, $L'$, on $E^*$. Moreover, we extend S{\l}odkowski joint spectra, $\sigma_{\delta,k}$, $\sigma_{\pi,k}$, and we study its usual spectral properties.