Banach空间上相容算子方程的最小范数解的扰动分析

Perturbation analysis for the minimal norm solution of the consistent operator equation in Banach spaces

设X,Y是Banach空间,T是D(T)СX到Y的稠定闭线性算子而且它的值域在Y闭.设相容算子方程Tx=b的非相容扰动为‖(T+δT)x-■‖=min■‖(T+δT)z-■‖,这里δT是X→Y的有界线性算子.在某些条件下(比如X,Y是自反的),设上述方程的最小范数解为■_m,并设Tx=b的解集S(T,b)中的最小范数解为x_m.本文给出了当δ(Ker T,Ker(T+δT))较小时,(dist(■_m,S(T,b))/‖X_m‖的上界估计式.

Let X, Y be Banach spaces and let T be a densely-defined closed linear operator from D（T） C to Y with closed range. Suppose the non-consistent perturbation of the consistent equation Tx = b is ｜｜（T＋δT）x-b^-｜=minz∈D（T）｜｜（t＋δT）z-b^-｜｜, where δT is a bounded linear operator from X to Y. Under certain conditions （e. g. X and Y are reflexive Banach spaces）, let x^-m be the minimal norm solution of above equation and let Xm be minimal norm solution of the set S（T, b） ={x ∈ D（T）｜ Tx = b}. In this paper, wegive an estimation of the upper bound of dist（x^-m,S（T,b））/｜｜xm｜｜ when δ（Ker T, Ker （T ＋ δT）） is small enough.