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. © 2006 Elsevier Inc. All rights reserved. MSC: 47B15; 17B20; 17B30