Finite-dimensional solvable Lie algebras generated by normal operators are commutative (Q2473804)

Let \(\mathcal H\) be a Hilbert space. The \(C^*\)-algebra of all bounded linear operators on \(\mathcal H\) is denoted by \(B(\mathcal H)\). On \(B(\mathcal H)\), one can define a Lie product \([T_1,T_2]:=T_1 T_2-T_2 T_1\) for \(T_1, T_2\in B(\mathcal H)\). For any subset \(M\subset B(\mathcal H)\), let \(\varepsilon(M)\subset B(\mathcal H)\) be the Lie subalgebra generated by \(M\). The main result of the paper is the following Theorem. If \(N_1, N_2\in B(\mathcal H)\) are normal operators, \(\dim \varepsilon(N_1, N_2)<\infty\), and \(\varepsilon(N_1, N_2)\) is solvable, then \(\varepsilon(N_1, N_2)\) is commutative. Some related results are given.