ON THE DISTRIBUTION OF THE DISCRETE SPECTRUM OF NUCLEARLY PERTURBED OPERATORS IN BANACH SPACES

Let Z(0) be a bounded operator in a Banach space X with purely essential spectrum and K a nuclear operator in X. We construct a holomorphic function the zeros of which coincide with the discrete spectrum of Z(0)+K and derive a Lieb-Thirring type inequality. We obtain estimates for the number of eigenvalues in certain regions of the complex plane and an estimate for the asymptotics of the eigenvalues approaching to the essential spectrum of Z(0).