Jacobson’s Lemma for Spectral Idempotents in Banach Algebras

Let a, b be elements in a complex Banach algebra with unity e. In this note, formulas are given for expressing the spectral idempotent of \(e - ba\) $e-ba$ associated with \(\sigma := \cup _{k = 1}^m\sigma _i\) $\sigma :={\cup }_{k=1}^m{\sigma }_i$ in terms of that of \(e - ab\) $e-ab$ , where \(\sigma _1, \sigma _2,..., \sigma _m\) ${\sigma }_1,{\sigma }_2,...,{\sigma }_m$ are finitely many pairwise disjoint spectral sets of \(e - ab\) $e-ab$ such that 1 is not in the convex hull of each \(\sigma _k\) ${\sigma }_k$ with \(1 \le k \le m\) $1\le k\le m$ . As an application, we establish the relation between the Drazin inverses of \(e - ab\) $e-ab$ and \(e - ba\) $e-ba$ both relative to \(\sigma \) $\sigma$ , which leads to a new way to recapture Jacobson’s lemma for the generalized Drazin inverse and generalized Drazin–Riesz inverse.