The stability radius of linear operator pencils

Abstract

Let T and S be two bounded linear operators from Banach spaces X into Y and suppose that T is Fredholm and the stability number k(T;S) is 0. Let d(T;S) be the supremum of all r > 0 such that dim N(T-λS) and codim R(T-λS) are constant for all λwith |λ| < r. It was proved in 1980 by H. Bart and D.C. Lay that d(T;S) = \lim_{n\to\infty}γ_{n}(T;S)^{1/n}, where γ_{n}(T;S) are some non-negative (extended) real numbers. For X=Y and S = I, the identity operator, we have γ_{n}(T;S) = γ(T^n), where γis the reduced minimum modulus. A different representation of the stability radius is obtained here in terms of the spectral radii of generalized inverses of T. The existence of generalized resolvents for Fredholm linear pencils is also considered.