Locally quasi-nilpotent elementary operators

Let $A$ be a unital dense algebra of linear mappings on a complex vector space $X$. Let $ϕ=\sum_{i=1}^n M_{a_i,b_i}$ be a locally quasi-nilpotent elementary operator of length $n$ on $A$. We show that, if $\{a_1,\ldots,a_n\}$ is locally linearly independent, then the local dimension of $V(ϕ)=\spa\{b_ia_j: 1 \leq i,j \leq n\}$ is at most $\frac{n(n-1)}{2}$. If $\lDim V(ϕ)=\frac{n(n-1)}{2} $, then there exists a representation of $ϕ$ as $ϕ=\sum_{i=1}^n M_{u_i,v_i}$ with $v_iu_j=0$ for $i\geq j$. Moreover, we give a complete characterization of locally quasi-nilpotent elementary operators of length 3.

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