Concrete Operators (Oct 2022)
m-isometric generalised derivations
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}.
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