Banach空间上相容算子方程的最小范数解的扰动分析

设~$X, Y$~是~Banach~空间, ~$T$~是 $mathcal{D}(T)subset X$~%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+delta T)x-barb|=minlimits_{zinmathcal{D}(T)}|(T+delta T)z-bar b|,$~%这里 $delta T$~是 $Xto Y$~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 $bar x_m$, 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$delta(Ker T, Ker(T+delta T))$~较小时, $dfrac{dist(bar x_m,S(T, b))}{|x_m|}$~的上界估计式.

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

Let~$X, Y$~ be Banach spaces and let $T$ be adensely--defined closed linear operator from $mathcal{D}(T)subset$to $Y$ with closed range. Suppose the non-consistent perturbationof the consistent equation $Tx=b$ is $ |(T+delta T)x-barb|=minlimits_{zinmathcal{D}(T)}|(T+delta T)z-bar b|, $where $delta T$ is a bounded linear operator from $X$ to $Y$. Undercertain conditions (e. g. $X$ and $Y$ are reflexive Banach spaces),let $bar x_m$ be the minimal norm solution of above equation andlet $x_m$ be minimal norm solution of the set $S(T,b)={xinmathcal{D}(T)vert, Tx=b}$. In this paper, we give anestimation of the upper bound of $dfrac{dist(bar x_m, S(T,b))}{|x_m|}$ when $delta(Ker T, Ker(T+delta T))$ is smallenough.