{"title":"m等轴张量积","authors":"Bhagawati Prashad Duggal, I. Kim","doi":"10.1515/conop-2022-0142","DOIUrl":null,"url":null,"abstract":"Abstract Given Banach space operators S i {S}_{i} and T i {T}_{i} , i = 1 , 2 i=1,2 , we use elementary properties of the left and right multiplication operators to prove, that if the tensor products pair ( S 1 ⊗ S 2 , T 1 ⊗ T 2 ) \\left({S}_{1}\\otimes {S}_{2},{T}_{1}\\otimes {T}_{2}) is strictly m m -isometric, i.e., Δ S 1 ⊗ S 2 , T 1 ⊗ T 2 m ( I ⊗ I ) = ∑ j = 0 m ( − 1 ) j m j ( S 1 ⊗ S 2 ) m − j ( T 1 ⊗ T 2 ) m − j = 0 {\\Delta }_{{S}_{1}\\otimes {S}_{2},{T}_{1}\\otimes {T}_{2}}^{m}\\left(I\\otimes I)={\\sum }_{j=0}^{m}{\\left(-1)}^{j}\\left(\\begin{array}{c}m\\\\ j\\end{array}\\right){\\left({S}_{1}\\otimes {S}_{2})}^{m-j}{\\left({T}_{1}\\otimes {T}_{2})}^{m-j}=0 , then there exist a non-zero scalar c c and positive integers m 1 , m 2 ≤ m {m}_{1},{m}_{2}\\le m such that m = m 1 + m 2 − 1 m={m}_{1}+{m}_{2}-1 , ( S 1 , c T 1 ) \\left({S}_{1},c{T}_{1}) is strict- m 1 {m}_{1} -isometric and S 2 , 1 c T 2 \\left({S}_{2},\\frac{1}{c}{T}_{2}\\right) is strict m 2 {m}_{2} -isometric.","PeriodicalId":53800,"journal":{"name":"Concrete Operators","volume":null,"pages":null},"PeriodicalIF":0.3000,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"m-Isometric tensor products\",\"authors\":\"Bhagawati Prashad Duggal, I. Kim\",\"doi\":\"10.1515/conop-2022-0142\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"Abstract Given Banach space operators S i {S}_{i} and T i {T}_{i} , i = 1 , 2 i=1,2 , we use elementary properties of the left and right multiplication operators to prove, that if the tensor products pair ( S 1 ⊗ S 2 , T 1 ⊗ T 2 ) \\\\left({S}_{1}\\\\otimes {S}_{2},{T}_{1}\\\\otimes {T}_{2}) is strictly m m -isometric, i.e., Δ S 1 ⊗ S 2 , T 1 ⊗ T 2 m ( I ⊗ I ) = ∑ j = 0 m ( − 1 ) j m j ( S 1 ⊗ S 2 ) m − j ( T 1 ⊗ T 2 ) m − j = 0 {\\\\Delta }_{{S}_{1}\\\\otimes {S}_{2},{T}_{1}\\\\otimes {T}_{2}}^{m}\\\\left(I\\\\otimes I)={\\\\sum }_{j=0}^{m}{\\\\left(-1)}^{j}\\\\left(\\\\begin{array}{c}m\\\\\\\\ j\\\\end{array}\\\\right){\\\\left({S}_{1}\\\\otimes {S}_{2})}^{m-j}{\\\\left({T}_{1}\\\\otimes {T}_{2})}^{m-j}=0 , then there exist a non-zero scalar c c and positive integers m 1 , m 2 ≤ m {m}_{1},{m}_{2}\\\\le m such that m = m 1 + m 2 − 1 m={m}_{1}+{m}_{2}-1 , ( S 1 , c T 1 ) \\\\left({S}_{1},c{T}_{1}) is strict- m 1 {m}_{1} -isometric and S 2 , 1 c T 2 \\\\left({S}_{2},\\\\frac{1}{c}{T}_{2}\\\\right) is strict m 2 {m}_{2} -isometric.\",\"PeriodicalId\":53800,\"journal\":{\"name\":\"Concrete Operators\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":0.3000,\"publicationDate\":\"2023-01-01\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Concrete Operators\",\"FirstCategoryId\":\"1085\",\"ListUrlMain\":\"https://doi.org/10.1515/conop-2022-0142\",\"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-2022-0142","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q4","JCRName":"MATHEMATICS","Score":null,"Total":0}
Abstract Given Banach space operators S i {S}_{i} and T i {T}_{i} , i = 1 , 2 i=1,2 , we use elementary properties of the left and right multiplication operators to prove, that if the tensor products pair ( S 1 ⊗ S 2 , T 1 ⊗ T 2 ) \left({S}_{1}\otimes {S}_{2},{T}_{1}\otimes {T}_{2}) is strictly m m -isometric, i.e., Δ S 1 ⊗ S 2 , T 1 ⊗ T 2 m ( I ⊗ I ) = ∑ j = 0 m ( − 1 ) j m j ( S 1 ⊗ S 2 ) m − j ( T 1 ⊗ T 2 ) m − j = 0 {\Delta }_{{S}_{1}\otimes {S}_{2},{T}_{1}\otimes {T}_{2}}^{m}\left(I\otimes I)={\sum }_{j=0}^{m}{\left(-1)}^{j}\left(\begin{array}{c}m\\ j\end{array}\right){\left({S}_{1}\otimes {S}_{2})}^{m-j}{\left({T}_{1}\otimes {T}_{2})}^{m-j}=0 , then there exist a non-zero scalar c c and positive integers m 1 , m 2 ≤ m {m}_{1},{m}_{2}\le m such that m = m 1 + m 2 − 1 m={m}_{1}+{m}_{2}-1 , ( S 1 , c T 1 ) \left({S}_{1},c{T}_{1}) is strict- m 1 {m}_{1} -isometric and S 2 , 1 c T 2 \left({S}_{2},\frac{1}{c}{T}_{2}\right) is strict m 2 {m}_{2} -isometric.