Perturbation analysis for the reduced minimum modulus of bounded linear operator in Banach spaces

摘要

Let X and Y be two Banach spaces over the complex field C and let T:X→Y be a bounded linear operator with the generalized inverse T+. Let T=T+δT be a bounded linear operator with ∥T+∥∥δT∥<1/2. Let γ(T) denote the reduced minimum modulus of T. We first establish the upper semi-continuity theorem for γ(T). Furthermore, if dim Ker T=dimKerT<+∞ or R(T)∩KerT+=0, then γ(T) is continuous under the operator norm. Also, we have a corollary in Hilbert spaces, which improves the result in [Linear Algebra Appl., 262 (1997) 229–242, Theorem 4.1].