Property (h) and Perturbations

 基金资助: 国家自然科学基金.  11561053国家自然科学基金.  11761029内蒙古自然科学基金.  2018BS01001内蒙古自治区高等学校科学研究项目.  NJZZ18018内蒙古自治区高等学校科学研究项目.  NJZY18021

 Fund supported: the NSFC.  11561053the NSFC.  11761029the Natural Science Foundation of Inner Mongolia.  2018BS01001the Research Program of Sciences at Universities of Inner Mongolia Autonomous Region.  NJZZ18018the Research Program of Sciences at Universities of Inner Mongolia Autonomous Region.  NJZY18021 Abstract

In this paper, we introduce and study the property (h), which extends a-Weyl's theorem. We consider its stability under commuting finite rank and nilpotent perturbations. We prove that property (h) on Banach spaces is related to an important property which has a leading role in local spectral theory:the single-valued extension property. From this result we deduce that property (h) holds for several classes of operators.

Keywords： Single-valued extension property ; a-Weyl's theorem ; Property (h)

Wurichaihu , Alatancang . Property (h) and Perturbations. Acta Mathematica Scientia[J], 2019, 39(4): 713-719 doi:

2 预备知识

$0 \in\sigma(T+F)\setminus\sigma_{SF_{+}^{-}}(T+F)$,则$T+F$是上半-Weyl算子,即

$T$满足广义$a$-Weyl定理,于是$T$满足$a$-Weyl定理.又$\sigma(T) = \sigma_a(T)$,故$T$满足$(h)$性质.

$F$是有限秩算子且

$T +F$不满足$a$-Weyl定理.从而$T+F$不满足$(h)$性质.

(ⅰ) $T$满足Weyl定理;

(ⅱ) $S$满足$(h)$性质.

因为$T\in L(X)$具有$(\beta)$性质,那么由文献可知$\sigma(T) = \sigma(S)$,故$iso\sigma(T) = iso\sigma(S)$.$(\beta)$性质蕴含$T$具有SVEP,再由文献的定理2.7可知$S$具有SVEP.由文献的定理5, $S$满足Weyl定理时$T$满足Weyl定理.因为$S$具有$(\delta)$性质可以推出$S^{*}$具有SVEP,再由定理3.3, (ⅰ)和(ⅱ)等价.

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