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 dim A (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.