Some equivalent conditions of stable perturbation of operators in Hilbert spaces

摘要

Let H1,H2 be two Hilbert spaces over the complex field C and let T,T=T+δT:H1→H2 be two bounded linear operators with the M–P generalized inverse T+. If R(T)∩R(T)⊥=0, we say that T is the stable perturbation of T. In this paper, we give five equivalent conditions that make T being the stable perturbation of T under the assumption ∥T+∥∥δT∥<1. These equivalent conditions generalize not only the notation of rank-preserving perturbations of matrices but also the notation of acute perturbations of matrices. As a result, we obtain the following:Suppose that R(T)∩R(T)⊥=0 and ∥T+∥∥δT∥<1. Then∥T+−T+∥⩽1+52∥T+∥∥T+∥∥δT∥.This result generalizes corresponding result when H1,H2 are all finite-dimensional.