Some Results on the Symmetric Representation of the Generalized Drazin Inverse in a Banach Algebra

Abstract

Based on the conditions a b 2 = 0 and b π ( a b ) ∈ A d , we derive that ( a b ) n , ( b a ) n , and a b + b a are all generalized Drazin invertible in a Banach algebra A , where n ∈ N and a and b are elements of A . By using these results, some results on the symmetry representations for the generalized Drazin inverse of a b + b a are given. We also consider that additive properties for the generalized Drazin inverse of the sum a + b .

Keywords

• generalized Drazin inverse
• Banach algebra
• representation