Cline’s formula and Jacobson’s lemma for g-Drazin inverse

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 ? A satisfying akbkak = ak+1 for some k ? N. We prove that a has g-Drazin inverse if and only if bkak has g-Drazin inverse. In this case, (bkak)d = bk(ad)2ak and ad = ak[(bkak)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.