  Let Z_0 be a bounded operator in a Banach space X with purely essential
spectrum and K a nuclear operator in X. Using methods of complex analysis we
study 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
