有限生成的算子李代数

Finitely Generated Lie Algebras of Operators

设A为结合代数,T1,…,Tn ∈A,ε(T)为T生成的李代数,这里记T= (T1,…,Tn)∈ An．本文证明:若ε(T)是幂零的,则其必为有限维的,并给出ε(T)为有限维幂零李代数的充要条件．然后考虑A为Banach空间X上有界线性算子组成的代数B(X),得到算子理论的一些结果;若拟幂零算子T1,T2生成的李代数是有限维幂零的,则T1T2,T1+T2均为拟幂零的．若非零紧算子T1与非标量算子T2生成的李代数是有限维的,则T2有非平凡超不变子空间,从而在形式上推广了有关不变子空间的 Lomonosov定理．

Let A be an associative algebra, T1,...,Tn∈ A. Write T - （T1,……,Tn）ε（T） denotes Lie algebra generated by T. It is shown that if ε（T） is a nilpotent Lie algebra, it is finite-dimensional. A sufficient and nccessary condition for ε（T） to be a finite-dimensional nilpotent Lie algebra is given. If A is considered as a Banach algcbra B（X） of all bounded lincar opcrators on a Banach space X, it is proved that if the Lie algebra generated by quasinilpotent T1, T2 is finite-dimensional nilpotent Lie algebra, T1T2, T1 ＋ T2 are also quasinilpotent; if the operator T1 and non-scalar value operator Lie algebra generated by non-zero compact T2 is finite-dimensional, T2 has nontrivial hyperinvariant subspace. This generalizes Lomonosov＇s theorem on invariant subspace of bounded linear operators.