# 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 $\phi=\sum {i=1}^n M {a i,b i}$ 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 $V\phi=\spa\{b ia j: 1 \leq i,j \leq n\}$ is at most $\frac{nn-1}{2}$. If $\lDim V\phi=\frac{nn-1}{2}$, then there exists a representation of $\phi$ as $\phi=\sum {i=1}^n M {u i,v i}$ with $v iu j=0$ for $i\geq j$. Moreover, we give a complete characterization of locally quasi-nilpotent elementary operators of length 3.