幂等元线性组合的Drazin可逆性

Drazin Invertibility of the Linear Combinations of Idempotents

设P,Q是有单位元的Banach代数U上的幂等元,a,b为非零复数.讨论了在PQP=PQ,PQP=QPQ,PQ=QP条件下,线性组合aP+bQ的Drazin逆表示和相应的Drazin指标,结果是相关文献定理的推广.特别地,利用线性组合aP+bQ的群逆的性质和表达式给出了P,Q可交换的一个充要条件.

Let P and Q be idempotents on the Banach algebra U with unit e,a and b be complex numbers with a≠0 and b≠0.The representations for the Drazin inverse of aP＋bQ under some condition such as PQP=PQ,PQP=QPQ and PQ=QP are obtained.Moreover,an necessary and sufficient conditions for PQ=QP holds is given.