Quasinilpotent operators in operator Lie algebras II
Volume 195 / 2009
Studia Mathematica 195 (2009), 193-200
MSC: Primary 47B07; Secondary 47L70.
DOI: 10.4064/sm195-2-6
Abstract
In this paper, it is proved that the Banach algebra $\overline{{\cal A}({\cal L})}$, generated by a Lie algebra $\cal L$ of operators, consists of quasinilpotent operators if $\cal L$ consists of quasinilpotent operators and $\overline{{\cal A}({\cal L})}$ consists of polynomially compact operators. It is also proved that $\overline{{\cal A}({\cal L})}$ consists of quasinilpotent operators if $\cal L$ 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.
