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 dimN(T − λS) is constant in a neighborhood of λ = 0. Let d(T ;S) be the supremum of all r > 0 such that dimN(T − λS) and codim R(T − λS) are constant for all λ with |λ| < r. It is a consequence of more general results due to H. Bart and D.C. Lay (1980) that d(T ;S) = limn→∞ γn(T ;S) , where γn(T ;S) are some non-negative (extended) real numbers. For X = Y and S = I, the identity operator, we have γn(T ;S) = γ(T ), where γ 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.