{"title":"抽象乘法算子的基本谱、范数和谱半径","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":"{\"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}","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
摘要
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。
The essential spectrum, norm, and spectral radius of abstract multiplication operators
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 .