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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.

A scattered element of a Banach algebra is an element with at most countable spectrum. The set of all scattered elements is denoted by (). The scattered radical ${}_{sc}\left(\right)$ is the largest ideal consisting of scattered elements. We characterize in several ways central elements of modulo the scattered radical. As a consequence, it is shown that the following conditions are equivalent: (i) () + () ⊂ (); (ii) ()() ⊂ (); (iii) $[\left(\right),]{\subset }_{sc}\left(\right)$.

It is proved that if $J_i$ is a Jordan operator on a Hilbert space with the Jordan decomposition $J_i=N_i+Q_i$, where $N_i$ is normal and $Q_i$ is compact and quasinilpotent, i = 1,2, and the Lie algebra generated by J₁,J₂ is an Engel Lie algebra, then the Banach algebra generated by J₁,J₂ is an Engel algebra. Some results for normal operators and Jordan operators on Banach spaces are given.