Quasinilpotent operators in operator Lie algebras (Q2455075)

The main results of the paper under review are related to the problem of finding infinite-dimensional versions of the classical theorem of Engel on matrix representations of finite-dimensional nilpotent Lie algebras. For instance, one of the theorems proved in the paper says that if \({\mathcal L}\) is a Lie algebra consisting of quasinilpotent operators on some Banach space such that the image of \({\mathcal L}\) in the corresponding Calkin algebra is a nilpotent Lie algebra, then every element in the norm-closed associative algebra \(\overline{{\mathcal A}({\mathcal L})}\) generated by \({\mathcal L}\) is again a quasinilpotent operator. By requiring \({\mathcal L}\) to be generated by a family of operators possessing rich spectral decompositions, one also obtains a condition ensuring that all of the operators in the image of the adjoint representation of \(\overline{{\mathcal A}({\mathcal L})}\) are quasinilpotent.