Continuity and General Perturbation of the Drazin Inverse for Closed Linear Operators

In this paper, we investigate a perturbation of the Drazin inverse AD of a closed linear operator A; the main tool for obtaining the estimates is the gap between subspaces and operators. By (X) we denote the set of all closed linear operators acting on a linear subspace of X to X , where X is a complex Banach space. We write (A), (A), (A), ρ(A), σ(A), and R(λ,A) for the domain, nullspace, range, resolvent set, spectrum, and the resolvent of an operator A ∈ (X). All relevant concepts from the theory of closed linear operators can be found in [3, 14]. The set of all operators T ∈ (X) with (T)= X will be denoted by (X); we recall that operators in (X) are bounded, and the operator norm of T ∈ (X) will be denoted by ‖T‖. An operator A ∈ (X) is called quasi-polar if 0 is a resolvent point or an isolated spectral point of A. The spectral projection of a quasi-polar operator A ∈ (X) is the unique idempotent operator Aπ ∈ (X) such that AAπx = AπAx for all x ∈ (A), AAπ is quasi-nilpotent, and A +Aπ is invertible in (X). With an eye to further development in this paper, we choose the following definition of the (generalized) Drazin inverse among the equivalent formulations given in [8].