广义幂等算子线性组合的稳定性定理

Stability Theorem for the Linear Combinations of Generalized Idempotents

设P,Q是Banach空间X上的两个广义幂等算子,满足Pm=P,Qn-1=Q,证明了当Pm-1Q=Qn-1P时,P,Q线性组合aP+bQ的群逆与非零复数a和b的选取无关,并给出相应群逆的表达式.

Let P and Q be generalized idempotents on the Banach space X such that Pm = P, Qn-1 = Q. It is proved that ifpm-1 Q = Qn-1 P, then the combination aP ＋ bQ of P and Q is group invertible for any given a ≠ 0 and b≠0 . Moreover, the representations for the group inverse of aP ＋ bQ are given.