Generalized Drazin Inverse Of The Product And Sum Of Operators And Drazin Inverse Of Anti-Triangular Operator Matrix

Generalized inverse theory is a very active research field of matrix theory and op-erator theory,and it has wide applications in numerical analysis,differential equations,numerical linear algebra,cybernetics.As a hot branch,Drazin inverse and its gener-alization have attracted the attention of scholars at home and abroad,and have been studied extensively.In this paper,we mainly study the generalized Drazin invertibility of the product and sum of two bounded linear operators and the Drazin invertibility of anti-triangular operator matrix.The details are as follows:(1)Based on the space decomposition method,we prove that the product PQ and the sum P+Q are generalized Drazin invertible under the conditions P3Q=PQP,Q3P=QPQ,PQ?P?=0,and the expressions of(PQ)d and(P+Q)d are given.(2)Under the conditions R(A)(?)R(B),R(I-B+B)(?)R(B),dimN(B)=1,we give the equivalent condition that the anti-triangular operator matrix(?)is Drazin invertible and ind(MI)?3.