{"title":"m-等距广义导子","authors":"B. Duggal, I. H. Kim","doi":"10.1515/conop-2022-0135","DOIUrl":null,"url":null,"abstract":"Abstract Given Banach space operators Ai, Bi (i = 1, 2), let δi denote (the generalised derivation) δi(X) = (LAi − RBi )(X) = AiX − XBi. If 0 ∈ σa(Bi), i = 1, 2, and if Δδ1,δ2n(I)=(Lδ1Rδ1-I)n(I)=0 \\Delta _{{\\delta _1},\\delta 2}^n\\left( I \\right) = {\\left( {{L_{{\\delta _1}}}{R_{{\\delta _1}}} - I} \\right)^n}\\left( I \\right) = 0 , then ΔA1,A2n(I)=0 \\Delta _{{A_1},A2}^n\\left( I \\right) = 0 . For Hilbert space pairs (A, B) such that 0 ∈ σa(B*) and Δδ*,δn(I)=0(i.e., δ is n-isometric) \\Delta _{{\\delta ^*},\\delta }^n\\left( I \\right) = 0\\left( {i.e.,\\,\\delta \\,is\\,n - isometric} \\right) , where δ= δA,B and δ* = δA* ,B*, this implies ΔA*,An(I)=0 \\Delta _{{A^*},A}^n\\left( I \\right) = 0 (and hence there exists a positivie integer m ≤ n such that A is strictly m-isometric). If Δδ*,δn(I)=0 \\Delta _{{\\delta ^*},\\delta }^n\\left( I \\right) = 0 , then there exists a scalar λ such that 0 ∈ σa((B − λI)*) and, given δ is strictly n-isometric, there exists a positive integer m ≤ n such that A − λI is strictly m-isometric. Furthermore, there exist decompositions ℋ = ℋ1 ⊕ ℋ2 and ℋ = ℋ11 ⊕ ℋ22 of ℋ and ti-nilpotent operators Ni (i = 1, 2) such that either A − λI = αI + N1 and B − λI = (0I|ℋ1 ⊕ 2eit I|ℋ2 ) + N2, or, A − λI = αI + N1, α = eit, 0 ≤ t < 2π, and B − λI = (0I|ℋ11 ⊕ 2eit I|ℋ22 ) + N2, or, A − λ = (α1I|ℋ1 ⊕ α2I|ℋ2 ) + N1 and Bλ= (0I|ℋ11 ⊕ μI|ℋ22 ) + N2, where μ = eit|μ|, 0 ≤ t < 2π, 0 < |μ| < 2, α1=eit| μ |+i4-| μ |22 {\\alpha _1} = {e^{it}}{{\\left| \\mu \\right| + i\\sqrt {4 - {{\\left| \\mu \\right|}^2}} } \\over 2} and α2=eit| μ |-i4-| μ |22 {\\alpha _2} = {e^{it}}{{\\left| \\mu \\right| - i\\sqrt {4 - {{\\left| \\mu \\right|}^2}} } \\over 2} .","PeriodicalId":53800,"journal":{"name":"Concrete Operators","volume":null,"pages":null},"PeriodicalIF":0.3000,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"m-isometric generalised derivations\",\"authors\":\"B. Duggal, I. H. Kim\",\"doi\":\"10.1515/conop-2022-0135\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"Abstract Given Banach space operators Ai, Bi (i = 1, 2), let δi denote (the generalised derivation) δi(X) = (LAi − RBi )(X) = AiX − XBi. If 0 ∈ σa(Bi), i = 1, 2, and if Δδ1,δ2n(I)=(Lδ1Rδ1-I)n(I)=0 \\\\Delta _{{\\\\delta _1},\\\\delta 2}^n\\\\left( I \\\\right) = {\\\\left( {{L_{{\\\\delta _1}}}{R_{{\\\\delta _1}}} - I} \\\\right)^n}\\\\left( I \\\\right) = 0 , then ΔA1,A2n(I)=0 \\\\Delta _{{A_1},A2}^n\\\\left( I \\\\right) = 0 . For Hilbert space pairs (A, B) such that 0 ∈ σa(B*) and Δδ*,δn(I)=0(i.e., δ is n-isometric) \\\\Delta _{{\\\\delta ^*},\\\\delta }^n\\\\left( I \\\\right) = 0\\\\left( {i.e.,\\\\,\\\\delta \\\\,is\\\\,n - isometric} \\\\right) , where δ= δA,B and δ* = δA* ,B*, this implies ΔA*,An(I)=0 \\\\Delta _{{A^*},A}^n\\\\left( I \\\\right) = 0 (and hence there exists a positivie integer m ≤ n such that A is strictly m-isometric). If Δδ*,δn(I)=0 \\\\Delta _{{\\\\delta ^*},\\\\delta }^n\\\\left( I \\\\right) = 0 , then there exists a scalar λ such that 0 ∈ σa((B − λI)*) and, given δ is strictly n-isometric, there exists a positive integer m ≤ n such that A − λI is strictly m-isometric. Furthermore, there exist decompositions ℋ = ℋ1 ⊕ ℋ2 and ℋ = ℋ11 ⊕ ℋ22 of ℋ and ti-nilpotent operators Ni (i = 1, 2) such that either A − λI = αI + N1 and B − λI = (0I|ℋ1 ⊕ 2eit I|ℋ2 ) + N2, or, A − λI = αI + N1, α = eit, 0 ≤ t < 2π, and B − λI = (0I|ℋ11 ⊕ 2eit I|ℋ22 ) + N2, or, A − λ = (α1I|ℋ1 ⊕ α2I|ℋ2 ) + N1 and Bλ= (0I|ℋ11 ⊕ μI|ℋ22 ) + N2, where μ = eit|μ|, 0 ≤ t < 2π, 0 < |μ| < 2, α1=eit| μ |+i4-| μ |22 {\\\\alpha _1} = {e^{it}}{{\\\\left| \\\\mu \\\\right| + i\\\\sqrt {4 - {{\\\\left| \\\\mu \\\\right|}^2}} } \\\\over 2} and α2=eit| μ |-i4-| μ |22 {\\\\alpha _2} = {e^{it}}{{\\\\left| \\\\mu \\\\right| - i\\\\sqrt {4 - {{\\\\left| \\\\mu \\\\right|}^2}} } \\\\over 2} .\",\"PeriodicalId\":53800,\"journal\":{\"name\":\"Concrete Operators\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":0.3000,\"publicationDate\":\"2022-01-01\",\"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-2022-0135\",\"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-2022-0135","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: 0
摘要
给定Banach空间算子Ai, Bi (i = 1,2),令δi表示(广义导数)δi(X) = (LAi−RBi)(X) = AiX−XBi。如果0∈σa(Bi), i = 1,2,如果Δδ1,δ2n(i) =(Lδ1Rδ1-I)n(i) =0 \Delta _{{\delta _1},\delta 2}^n\left(1) \right) = {\left( {{我……{{\delta _1}}}{r_{{\delta _1}}} -我} \right)^n}\left(1) \right)=0,则ΔA1,A2n(I)=0 \Delta _{{a_1}, a}^n\left(1) \right) = 0。对于希尔伯特空间对(A, B),使得0∈σa(B*)和Δδ*,δn(I)=0(即δ是n等距) \Delta _{{\delta ^*},\delta }^n\left(1) \right) = 0\left( {即,\,\delta \,是\,n -等距} \right),其中δ= δ a,B和δ* = δ a *, B*,这意味着ΔA*,An(I)=0 \Delta _{{a ^*}, a}^n\left(1) \right) = 0(因此存在一个正整数m≤n,使得a是严格m等距的)。如果Δδ*,δn(I)=0 \Delta _{{\delta ^*},\delta }^n\left(1) \right) = 0,则存在一个标量λ使得0∈σa((B−λ i)*),并且,给定δ是严格n等距的,则存在一个正整数m≤n使得a−λ i是严格m等距的。进一步地,存在着分解h = h = h 1⊕h 2和h = h 11⊕h 22的h和反幂零算子Ni (i = 1,2),使得A−λ i = α i + N1和B−λ i = (0I| h 1⊕2eit i | h 2) + N2,或者A−λ= (α i + N1, α =eit, 0≤t < 2π, B−λ i = (0I| h 11⊕2eit i | h 22) + N2,或者A−λ= (α 1i | h 1⊕α 2i | h 2) + N1和Bλ= (0I| h 11⊕μ i | h 22) + N2,其中μ =eit| μ|, 0≤t < 2π, 0 < |μ| < 2, α1=eit| μ| +i4-| μ| 22 {\alpha _1} = {e^{它}}{{\left| \mu \right| + I\sqrt {4 - {{\left| \mu \right|}^2}} } \over 2} α2=eit| μ |-i4-| μ |22 {\alpha _2} = {e^{它}}{{\left| \mu \right| - I\sqrt {4 - {{\left| \mu \right|}^2}} } \over 2} .
Abstract Given Banach space operators Ai, Bi (i = 1, 2), let δi denote (the generalised derivation) δi(X) = (LAi − RBi )(X) = AiX − XBi. If 0 ∈ σa(Bi), i = 1, 2, and if Δδ1,δ2n(I)=(Lδ1Rδ1-I)n(I)=0 \Delta _{{\delta _1},\delta 2}^n\left( I \right) = {\left( {{L_{{\delta _1}}}{R_{{\delta _1}}} - I} \right)^n}\left( I \right) = 0 , then ΔA1,A2n(I)=0 \Delta _{{A_1},A2}^n\left( I \right) = 0 . For Hilbert space pairs (A, B) such that 0 ∈ σa(B*) and Δδ*,δn(I)=0(i.e., δ is n-isometric) \Delta _{{\delta ^*},\delta }^n\left( I \right) = 0\left( {i.e.,\,\delta \,is\,n - isometric} \right) , where δ= δA,B and δ* = δA* ,B*, this implies ΔA*,An(I)=0 \Delta _{{A^*},A}^n\left( I \right) = 0 (and hence there exists a positivie integer m ≤ n such that A is strictly m-isometric). If Δδ*,δn(I)=0 \Delta _{{\delta ^*},\delta }^n\left( I \right) = 0 , then there exists a scalar λ such that 0 ∈ σa((B − λI)*) and, given δ is strictly n-isometric, there exists a positive integer m ≤ n such that A − λI is strictly m-isometric. Furthermore, there exist decompositions ℋ = ℋ1 ⊕ ℋ2 and ℋ = ℋ11 ⊕ ℋ22 of ℋ and ti-nilpotent operators Ni (i = 1, 2) such that either A − λI = αI + N1 and B − λI = (0I|ℋ1 ⊕ 2eit I|ℋ2 ) + N2, or, A − λI = αI + N1, α = eit, 0 ≤ t < 2π, and B − λI = (0I|ℋ11 ⊕ 2eit I|ℋ22 ) + N2, or, A − λ = (α1I|ℋ1 ⊕ α2I|ℋ2 ) + N1 and Bλ= (0I|ℋ11 ⊕ μI|ℋ22 ) + N2, where μ = eit|μ|, 0 ≤ t < 2π, 0 < |μ| < 2, α1=eit| μ |+i4-| μ |22 {\alpha _1} = {e^{it}}{{\left| \mu \right| + i\sqrt {4 - {{\left| \mu \right|}^2}} } \over 2} and α2=eit| μ |-i4-| μ |22 {\alpha _2} = {e^{it}}{{\left| \mu \right| - i\sqrt {4 - {{\left| \mu \right|}^2}} } \over 2} .