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A note on the generalized Kato spectrum of an operator
JIANG Qiao-Fen
School of Mathematics and Computer Science , Fujian Normal University, Fuzhou 350108
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Opened online: 2 December 2015
Accepted by: none
Citation: JIANG Qiao-Fen.A note on the generalized Kato spectrum of an operator[OL]. [ 2 December 2015] http://en.paper.edu.cn/en_releasepaper/content/4665731
 
 
Let $X$ be a Banach space and $T$ be a bounded linear operator on $X$. We denote by $mathcal{S}(T)$ the set of all complex $lambdain mathbb{C}$ such that $T$ does not have the single-valued extension property at $lambda$. In this note, we prove equality up to $mathcal{S}(T)$ between the appoximate point spectrum and the generalized Kato spectrum, equality up to $mathcal{S}(T^*)$ between the surjectivity spectrum and the generalized Kato spectrum.We give some applications of these results on the commuted quasi-nilpotent perturbation of operators with generalized Kato decomposition and the generalized Kato spectrum of the operator matrices, we also discuss the generalized Kato spectrum of the multipliers on the direct sum of Banach algebras.
Keywords:fundamental mathematics;Banach space; generalized Kato decomposition; generalized Drazin invertible; single-valued extension property
 
 
 

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