The Stability Radius of Fredholm Linear Pencils

Let T and S be two bounded linear operators from Banach spaces X into Y, and suppose that T is Fredholm and dim NŽT S.is constant in a neighborhood of 0. Let dŽT; S.be the supremum of all r 0 such that dim NŽT S.and codim RŽT S.are constant for all with r. It is a consequence of more general results due to H. Bart and D. C. LayŽ1980, Studia Math. 66, 307 320.that dŽT; S. lim n nŽT; S.1 n, where nŽT; S.are some non-negativeŽextended. real numbers. For X Y and S I, the identity operator, we have nŽT; S. ŽT n., where is the reduced minimum modulus. A different representation of the stability radius dŽT; 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.