摘要
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.
出处
《数学学报(中文版)》
SCIE
CSCD
北大核心
2006年第1期45-50,共6页
Acta Mathematica Sinica:Chinese Series
基金
国家自然科学基金资助项目(10271032)
关键词
幂零李代数
拟幂零算子
不变子空间
nilpotent operator
quasinilpotent operator
invariant subspace