The stability radius of Fredholm linear pencils

Let T and S be two bounded linear operators from Banach spaces X into Y, an d suppose that T is Fredholm and dim N(T - lambdaS) is constant in a neighb orhood of lambda = 0. Let d(T; S) be the supremum of all r > 0 such that di m N(T - lambdaS) and codim R(T - lambdaS) are constant for all lambda with \ lambda \ < r. It is a consequence of more general results due to H. Bart and D. C. Lay (1980, Studia Math. 66, 307-320) that d(T; S) = lim(n --> inf inity)gamma (n)(T; S)(1/n), where gamma (n)(T; S) are some non-negative (ex tended) real numbers. For X = Y and S = I, the identity operator, we have g amma (n)(T; S) = gamma (T-n), where gamma is the reduced minimum modulus. A different representation of the stability radius d(T; S) 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. (C) 2001 Academic Press.