Journal of East China Normal University(Natural Sc ›› 2009, Vol. 2009 ›› Issue (1): 48-52.

• Article • Previous Articles     Next Articles

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

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

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

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