有限生成算子李代数的几个结果

Some Results for Lie Algebra of the Finitely Generated Operators

主要研究有限生成算子李代数的几个重要结果．通过设A为结合代数，T1…，Ln∈A，ε（T）为T生成的李代数，这里记T=（T1，…，Tn）∈A^n，讨论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T1＋T2，T1T2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．

In order to study some results for Lie algebra of the finitely generated operators, let A be an associative algebra and be T1…,Ln∈A, Setting T = （ T1,…,Tn）, and ε （T） denotes Lie algebra generated by T. Suppose that A is a Banach algebra B （X） of all the bounded linear operators on a Banach space X. In this paper, the author discussed A and got some theoretical results. That is, if the Lie algebra generated by quasi-nilpotent operators T1, T2 is finite-dimensional nilpotent Lie algebra, then T1 T2,T1 ＋ T2 are also quasi-nilpotent; if the Lie algebra generated by non-zero compact operator T1 and nonscalar value operator T2 is finite-dimensional, then T2 has non-trivial hyperinvariant subspace. Thus they had generalized in form Lomonosov＇ s theorem on invariant subspace of bounded linear operators.