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 $L$, $L^{\mathrm{\prime }}$, on $E^{\ast }$. Moreover, we extend S{\l}odkowski joint spectra, ${\sigma }_{\delta ,k}$, ${\sigma }_{\pi ,k}$, and we study its usual spectral properties.