Cline's Formula and Jacobson's Lemma for G-Drazin inverse

Abstract

We present new conditions under which Cline’s
formula and Jacobson’s lemma for g-Drazin inverse hold. Let
A be a Banach algebra, and let a; b\in A satisfying a^kb^ka^k =
a^{k+1} for some k \in N. We prove that a has g-Drazin inverse if
and only if bkak has g-Drazin inverse. In this case,
(b^ka^k)^d = b^k(a^d)^2a^k and a^d = a^k[(b^k^ak)^d]^{k+1}:
Further, we study Jacobson’s lemma for g-Drazin inverse in a
Banach algebra under the preceding condition. The common
spectral property of bounded linear operators on a Banach
space is thereby obtained.