Lie algebras generated by bounded linear operators on Hilbert spaces

It is proved that the operator Lie algebra ε(T,T∗) generated by a bounded linear operator T on Hilbert space H is finite-dimensional if and only if T=N+Q, N is a normal operator, [N,Q]=0, and dimA(Q,Q∗)<+∞, where ε(T,T∗) denotes the smallest Lie algebra containing T,T∗, and A(Q,Q∗) denotes the associative subalgebra of B(H) generated by Q,Q∗. Moreover, we also give a sufficient and necessary condition for operators to generate finite-dimensional semi-simple Lie algebras. Finally, we prove that if ε(T,T∗) is an ad-compact E-solvable Lie algebra, then T is a normal operator.