m-isometric generalised derivations

IF 0.3 Q4 MATHEMATICS
B. Duggal, I. H. Kim
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引用次数: 0

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} .
m-等距广义导子
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} .
本文章由计算机程序翻译,如有差异,请以英文原文为准。
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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
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