Banach代数中上三角矩阵的广义Drazin谱

Generalized Drazin Spectrum of Upper Triangular Matrices in Banach Algebras

设A是含单位元e的Banach代数,a,b,c∈A,M_(c)=(a c o b)∈M_(2)(A).本文提出了 Banach代数中元素的左、右广义Drazin可逆的概念.定义集合σgD(a)={λ∈C:a - λe不是广义Drazin可逆的}为元素a的广义Drazin谱.证明了σgD(a)∪σgD(a)=σgD(M_(c))∪W_(g),其中 W_(g) 是σgD (M_(c))的某些洞且W_(g)■σgD(a)∩σgD(b),或者更精细地W_(g)■σrgD(a)∩σlgD(b).此外,还研究了 Banach代数中元素的广义Drazin谱的其他性质.

Let A be a Banach algebra with unit e and a,b,c∈A,M_(c)=(a c o b)∈M_(2)(A).The concepts of left and right generalized Drazin invertible of elements in a Banach algebra are proposed.A generalized Drazin spectrum of a is defined by σgD(a)={λ∈C:a-λe is not generalized Drazin invertible}.It is shown thatσgD(a)∪σgD(b)=σgD(Mc)∪W_(g),where W_(g) is a union of certain holes in σgD(M_(c)) and W_(g)■σgD(a)∩σgD(b),or more finely W_(g)■σrgD(a)∩σlgD(b).In addition,some properties of generalized Drazin spectrum of elements in a Banach algebra are studied.