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-\lambda S) and codim R(T-\lambda S) are constant for all \lambda with |\lambda | < r. It was proved in 1980 by H. Bart and D.C. Lay that d(T;S) = \lim_{n\to\infty}\gamma_{n}(T;S)^{1/n}, where \gamma_{n}(T;S) are some non-negative (extended) real numbers. For X=Y and S = I, the identity operator, we have \gamma_{n}(T;S) = \gamma (T^n), where \gamma is 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.