Banach空间中线性算子的Drazin逆

借助线性算子的von Neumann正则逆，给出了Banach空间中线性算子的Drazin逆的一个判别准则及表达式，即：设A为Banach空间X上的线性算子，k为正整数，如果A^k有von Neumann正则逆（A^k)^(1)，则A有（1^k,2,5)－逆（即为A^D）当且仅当U＝A^k 1(A^k)^(1)＋I－A^k(A^k)^(1）可逆当且仅当V＝（A^k)^(1)A^k 1 I-(A^k)^(1)A^k可逆，且此时，A^D=U^-(k 1)A^k=A^kV^-(k 1)=U^-1A^kV^-k,从而推广了Puystjens和Hartwig关于群逆的一个结果。

The Drazin Inverse of a Linear Operator over a Banach Space

By using of von Neumann regular inverses of linear operators, we give a characterization and a formulae for the Drazin inverse of a linear operator over a Banach space, that is: let A be a linear operator over a Banach space and k a nature number. If A4 has a von Neumann regular inverse (Ak)(1), then A has a (1 ,2,5) -inverse (i. e. AD) if and only if U = Ak+1 (Ak)(1) + I - A(Ak)(1) is invertible if and only if V= (Ak)(1) Ak+1 +I- (Ak)(1)A is invertible . In this case, AD= U-(k+1)Ak=AkV-(k+1) = U-AkV-1 = U-1AV-k. This result generalizes a result for the group inverse of Puystjens and Hartwig.