We prove in particular that Banach spaces of the form C₀(Ω), where Ω is a locally compact space, enjoy a quantitative version of the reciprocal Dunford-Pettis property.

In this paper, it is proved that the Banach algebra $\overline{\left(ℒ\right)}$, generated by a Lie algebra ℒ of operators, consists of quasinilpotent operators if ℒ consists of quasinilpotent operators and $\overline{\left(ℒ\right)}$ consists of polynomially compact operators. It is also proved that $\overline{\left(ℒ\right)}$ consists of quasinilpotent operators if ℒ is an essentially nilpotent Engel Lie algebra generated by quasinilpotent operators. Finally, Banach algebras generated by essentially nilpotent Lie algebras are shown to be compactly quasinilpotent.