华东师范大学学报(自然科学版) ›› 2009, Vol. 2009 ›› Issue (1): 48-52.

• 应用数学与基础数学 • 上一篇    下一篇

Banach空间上相容算子方程的最小范数解的扰动分析

汪晶晶, 黎志华, 薛以锋   

  1. 华东师范大学,数学系, 上海200241
  • 收稿日期:2008-04-10 修回日期:2008-05-18 出版日期:2009-01-25 发布日期:2009-01-25
  • 通讯作者: 薛以锋

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

WANG Jing-jing, LI Zhi-hua, XUE Yi-feng

  

  1. Department of Mathematics, East China Normal University, Shanghai 200241, China
  • Received:2008-04-10 Revised:2008-05-18 Online:2009-01-25 Published:2009-01-25
  • Contact: XUE Yi-feng

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摘要： 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-\bar
b\|=\min\limits_{z\in\mathcal{D}(T)}\|(T+\delta T)z-\bar b\|,
$~%
这里\ $\delta T$~是\ $X\to Y$~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.

关键词: 闭值域, 约化最小模, 最小范数解, 闭值域, 约化最小模, 最小范数解

Abstract: Let~$X, Y$~ be Banach spaces and let $T$ be a
densely--defined closed linear operator from $\mathcal{D}(T)\subset$
to $Y$ with closed range. Suppose the non-consistent perturbation
of the consistent equation $Tx=b$ is $ \|(T+\delta T)x-\bar
b\|=\min\limits_{z\in\mathcal{D}(T)}\|(T+\delta T)z-\bar b\|, $
where $\delta T$ is a bounded linear operator from $X$ to $Y$. Under
certain conditions (e. g. $X$ and $Y$ are reflexive Banach spaces),
let $\bar x_m$ be the minimal norm solution of above equation and
let $x_m$ be minimal norm solution of the set $S(T,
b)=\{x\in\mathcal{D}(T)\vert\, Tx=b\}$. In this paper, we give an
estimation of the upper bound of $\dfrac{\dist(\bar x_m, S(T,
b))}{\|x_m\|}$ when $\delta(\Ker T, \Ker(T+\delta T))$ is small
enough.

Key words: reduced minimum modulus, minimal norm solution, closed range, reduced minimum modulus, minimal norm solution

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