{"title":"The essential spectrum, norm, and spectral radius of abstract multiplication operators","authors":"A. R. Schep","doi":"10.1515/conop-2022-0141","DOIUrl":null,"url":null,"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 .","PeriodicalId":53800,"journal":{"name":"Concrete Operators","volume":null,"pages":null},"PeriodicalIF":0.3000,"publicationDate":"2022-04-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"1","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Concrete Operators","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.1515/conop-2022-0141","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q4","JCRName":"MATHEMATICS","Score":null,"Total":0}
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 .