Perturbation Analysis for the Operator Equation Txsb in Banach Spaces

Let X1, X2 be two Banach spaces over the complex field C and let T: X1ªX2 be a bounded linear operator with the generalized inverse Tq. Let TsTqd T be a bounded linear operator with 5Tq5 5d T 5-1. Suppose that dimker Ts dimker T-` or R T.lKer Tqs0. Then T has the generalized inverse Tqs IqTqd T.y1Tq with 5Tq5 5Tq5F . 1y5Tq5 5d T 5 This result generalizes Theorem 3.9 of M. Z. Nashed ‘‘Generalized Inverses and Applications,’’ Academic Press, New York, 1976.. Using this result, we give some results of the perturbation analysis for the operator equation Txsb.