Additive property for generalized Zhou inverse in a Banach algebra

Abstract

Let A be a Banach Algebra. An element a \in A has generalized Zhou inverse if there exists b \in A  such that
b = bab; ab = ba; a^n - ab \in J^#(A); for some n \in N. We find some new conditions under which the generalized Zhou inverse of the sum a + b can be explicity expressed in terms of a; b; a^z; b^z. In particular, necessary and sufficient conditions for the existence of the generalized Zhou inverse of the sum a + b are obtained.