Some equivalent conditions of stable perturbation of operators in Hilbert spaces

摘要

Let H-1, H-2 be two Hilbert spaces over the complex field C and let T, (T) over bar = T + deltaT : H-1 --> H-2 be two bounded linear operators with the M-P generalized inverse T+. If R((T) over bar) boolean AND R(T)(perpendicular to) = 0, we say that (T) over bar 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+ deltaT < 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) over bar) boolean AND R(T)(perpendicular to) = 0 and T+ deltaT < 1. Then (T) over bar (+) -T+ less than or equal to 1 + root5/2 (T) over bar (+) less than or equal to 1 + root5/2 (t) over bar (+)deltaT. This result generalizes corresponding result when H-1, H-2 are all finite-dimensional. (C) 2003 Elsevier Inc. All rights reserved. [References: 9]