A note on common properties for some generalized inverses of elements in banach algebra

Let A be a Banach algebra, and let a,b is an element of A satisfying a(k)b(k)a(k) = a(k+1) for some k is an element of N. Under this condition, we show that Cline's formula holds for some new generalized inverses, so a has strong Drazin inverse (respectively, generalized strong Drazin inverse, Hirano inverse and generalized Hirano inverse) if and only if b(k)a(k) has strong Drazin inverse (respectively, generalized strong Drazin inverse, Hirano inverse and generalized Hirano inverse). As a particular case, some results for Jacobson's lemma are given.