| Abstract:
The ordinary inverse in rings or algebras does not behave well with respect
to addition. If \( x \) and \( y \) are invertible elements in a unital
ring, we cannot expect that the sum \( x+y \) is invertible, and even less
that the inverse of this sum is the sum of inverses (under some conditions
on \( x \) and \( y \)). It therefore comes as a surprise that the
Drazin inverse (as well as its generalized version) is quite
amenable under addition. In this talk, we shall study some additive
properties of the generalized Drazin inverse in a Banach algebra and find
explicit expressions for the generalized Drazin inverse of the sum
\( x+y \) in terms of \( x \) and \( y \) and their generalized
Drazin inverses under fairly mild conditions on \( x \) and \( y \).
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