A generalization of Banach’s lemma and its applications to perturbations of bounded linear operators

Let X be a Banach space and let P : X → X be a bounded linear operator. Using an algebraic inequality on the spectrum of P , we give a new sufficient condition that guarantees the existence of ( I – P ) −1 as a bounded linear operator on X , and a bound on its spectral radius is also obtained. This generalizes the classic Banach lemma. We apply the result to the perturbation analysis of general bounded linear operators on X with commutative perturbations.