Locally quasi-nilpotent elementary operators

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Let A be a unital dense algebra of linear mappings on a complex vector space X. Let ϕ=∑ᵢ₌₁ⁿ Mₐᵢ,bᵢ be a locally quasi-nilpotent elementary operator of length n on A. We show that, if {a₁,,aₙ} is locally linearly independent, then the local dimension of V(ϕ)={bᵢaⱼ: 1 ≤ i,j ≤ n} is at most n(n-1)/2. If V(ϕ)=n(n-1)/2, then there exists a representation of ϕ as ϕ=∑ᵢ₌₁ⁿ Mᵤᵢ,ᵥᵢ with vᵢuⱼ=0 for i≥ j. Moreover, we give a complete characterization of locally quasi-nilpotent elementary operators of length 3.