m-isometric generalised derivations

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 ∈ σ a (Bi ), i = 1, 2, and if Δ δ 1 , δ 2 n ( I ) = ( L δ 1 R δ 1 - 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 Δ A 1 , A 2 n ( I ) = 0 \Delta _{{A_1},A2}^n\left( I \right) = 0 . For Hilbert space pairs (A, B) such that 0 ∈ σ a (B *) and Δ δ * , δ n ( I ) = 0 ( i . e . ,   δ   i s   n - i s o m e t r i c ) \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 Δ A * , A n ( 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 Δ δ * , δ n ( I ) = 0 \Delta _{{\delta ^*},\delta }^n\left( I \right) = 0 , then there exists a scalar λ such that 0 ∈ σ a ((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 + N 1 and B − λI = (0I|ℋ1 ⊕ 2eit I|ℋ2 ) + N 2, or, A − λI = αI + N 1, α = eit , 0 ≤ t < 2π, and B − λI = (0I|ℋ11 ⊕ 2eit I|ℋ22 ) + N 2, or, A − λ = (α 1 I|ℋ1 ⊕ α 2 I|ℋ2 ) + N 1 and Bλ = (0I|ℋ11 ⊕ μI|ℋ22 ) + N 2, where μ = eit |μ|, 0 ≤ t < 2π, 0 < |μ| < 2, α 1 = e i t | μ | + i 4 - | μ | 2 2 {\alpha _1} = {e^{it}}{{\left| \mu \right| + i\sqrt {4 - {{\left| \mu \right|}^2}} } \over 2} and α 2 = e i t | μ | - i 4 - | μ | 2 2 {\alpha _2} = {e^{it}}{{\left| \mu \right| - i\sqrt {4 - {{\left| \mu \right|}^2}} } \over 2} .