Explicit characterization of the Drazin index

Let B(X) be the set of bounded linear operators on a Banach space X, and A∈B(X) be Drazin invertible. An element B∈B(X) is said to be a stable perturbation of A if B is Drazin invertible and I-Aπ-Bπ is invertible, where I is the identity operator on X, Aπ and Bπ are the spectral projectors of A and B, respectively. Under the condition that B is a stable perturbation of A, a formula for the Drazin inverse BD is derived. Based on this formula, a new approach is provided to the computation of the explicit Drazin indices of certain 2×2 operator matrices.