Lie algebras generated by bounded linear operators on Hilbert spaces (Q860585)

Let \(H\) be a Hilbert space and let \(T\in B(H)\). It is proved that the Lie algebra generated by \(T\) and \(T^*\) is finite-dimensional if and only if \(T=N+Q\), where \(N\) is a normal operator, \([N,Q]=0\), and the Lie algebra generated by \(Q\) and \(Q^*\) is finite-dimensional and semisimple (and hence the associative algebra generated by \(Q\) and \(Q^*\) is finite-dimensional, too). Operators \(Q\) such that the Lie algebra generated by \(Q\) and \(Q^*\) is finite-dimensional and semisimple are also characterized. Finally, it is proved that if the Lie algebra generated by \(T\) and \(T^*\) is ad-compact and E-solvable, then \(T\) is a normal operator.