The essential spectrum, norm, and spectral radius of abstract multiplication operators

IF 0.3 Q4 MATHEMATICS Concrete Operators Pub Date : 2022-04-08 DOI:10.1515/conop-2022-0141
A. R. Schep
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引用次数: 1

Abstract

Abstract Let E E be a complex Banach lattice and T T is an operator in the center Z ( E ) = { T : ∣ T ∣ ≤ λ I for some λ } Z\left(E)=\left\{T:| T| \le \lambda I\hspace{0.33em}\hspace{0.1em}\text{for some}\hspace{0.1em}\hspace{0.33em}\lambda \right\} of E E . Then, the essential norm ‖ T ‖ e \Vert T{\Vert }_{e} of T T equals the essential spectral radius r e ( T ) {r}_{e}\left(T) of T T . We also prove r e ( T ) = max { ‖ T A d ‖ , r e ( T A ) } {r}_{e}\left(T)=\max \left\{\Vert {T}_{}\hspace{-0.35em}{}_{{A}^{d}}\Vert ,{r}_{e}\left({T}_{A})\right\} , where T A {T}_{A} is the atomic part of T T and T A d {T}_{}\hspace{-0.35em}{}_{{A}^{d}} is the nonatomic part of T T . Moreover, r e ( T A ) = limsup ℱ λ a {r}_{e}\left({T}_{A})={\mathrm{limsup}}_{{\mathcal{ {\mathcal F} }}}{\lambda }_{a} , where ℱ {\mathcal{ {\mathcal F} }} is the Fréchet filter on the set A A of all positive atoms in E E of norm one and λ a {\lambda }_{a} is given by T A a = λ a a {T}_{A}a={\lambda }_{a}a for all a ∈ A a\in A .
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抽象乘法算子的基本谱、范数和谱半径
设E E为复巴拿赫格,T T为中心Z (E) =上的算子 { T:∣T∣≤λ I对于某些λ } z\left(e)=\left{t:| t | \le \lambda I\hspace{0.33em}\hspace{0.1em}\text{for some}\hspace{0.1em}\hspace{0.33em}\lambda \right} E E。然后,基本规范‖T‖e \Vert t{\Vert }_{e} T =本质谱半径r e (T) {r}_{e}\left(T) (T)我们也证明了re (T) = max { ‖T A d‖,r e (T A) } {r}_{e}\left(t)=\max \left{\Vert {t}_{}\hspace{-0.35em}{}_{{a}^{d}}\Vert ,{r}_{e}\left({t}_{a})\right},其中T {t}_{a} T的原子部分是T还是T是d {t}_{}\hspace{-0.35em}{}_{{a}^{d}} 是T T的非原子部分。并且,re (T A) = limsup λ A {r}_{e}\left({t}_{a})={\mathrm{limsup}}_{{\mathcal{ {\mathcal F} }}}{\lambda }_{a} ,其中: {\mathcal{ {\mathcal F} }} fr过滤器是在集合A A上的吗?集合A A是E E的范数1和λ A的所有正原子 {\lambda }_{a} 由T A A = λ A A给出 {t}_{a}a={\lambda }_{a}对于所有的a∈a\in 选A。
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来源期刊
Concrete Operators
Concrete Operators MATHEMATICS-
CiteScore
1.00
自引率
16.70%
发文量
10
审稿时长
22 weeks
期刊最新文献
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