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May 2007 Perturbation Analysis for the Drazin Inverse Under Stable Perturbation in Banach Space
Yifeng Xue, Guoliang Chen
Missouri J. Math. Sci. 19(2): 106-120 (May 2007). DOI: 10.35834/mjms/1316092490

Abstract

Let $X$ be Banach space and let $T, \bar T=T+\delta T$ be bounded linear operators on $X$. Suppose that $T$ has the Drazin inverse $T^D$ and $\text{Ind} (T)=n$. In this paper, we show that if $\|\delta T\|$ is sufficiently small and $\text{Ran} (\bar T^n) \cap \text{Ker} (({T^D})^n) = \{0\}$, then $\bar T$ is Drazin invertible with $\text{Ind} (\bar T)\le n$. In this case, the expression of $\bar T^D$ is given and the upper bounds of $\|\bar T^D\|$ and $$\dfrac{\|\bar T^D-T^D\|}{\|T^D\|}$$ are established. If $\dim X<\infty$, replacing $\text{Ran} (\bar T^n) \cap \text{Ker} (({T^D})^n) = \{0\}$ by $\text{rank} (\bar T^n) = \text{rank} (T^n)$, we obtain the same perturbation results of the Drazin invertible matrix $T$ as in the case of $\dim X=\infty$.

Citation

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Yifeng Xue. Guoliang Chen. "Perturbation Analysis for the Drazin Inverse Under Stable Perturbation in Banach Space." Missouri J. Math. Sci. 19 (2) 106 - 120, May 2007. https://doi.org/10.35834/mjms/1316092490

Information

Published: May 2007
First available in Project Euclid: 15 September 2011

zbMATH: 1169.47004
Digital Object Identifier: 10.35834/mjms/1316092490

Subjects:
Primary: 47A05
Secondary: 65F20 , 65J1

Rights: Copyright © 2007 Central Missouri State University, Department of Mathematics and Computer Science

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Vol.19 • No. 2 • May 2007
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