Quasinilpotent operators in operator Lie algebras IV

In this paper, we show that if a Lie algebra L \mathcal {L} is of scattered operators and the norm closure of [ L , L ] [\mathcal {L},\mathcal {L}] is of polynomially compact operators, then the closed algebra generated by L \mathcal {L} is commutative modulo the Jacobson radical if and only if [ L , L ] [\mathcal {L},\mathcal {L}] is of quasinilpotent operators. This gives a partial answer of Wojtyński’s problem. We also show that an essentially nilpotent Lie algebra L \mathcal {L} generated by scattered operators is Engel if and only if the norm closed algebra generated by L \mathcal {L} is Engel, which gives a partial answer for a question posed by Shul′man and Turovskiĭ on Engel algebras.