Locally quasi-nilpotent elementary operators

Abstract

Let $A$ be a unital dense algebra of linear mappings on a complex vector space $X$. Let $\varphi ={\sum }_{i=1}^n{M}_{a_i,b_i}$ be a locally quasi-nilpotent elementary operator of length $n$ on $A$. We show that, if $\{a_1,\dots ,a_n\}$ 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 $\varphi$ as $\varphi ={\sum }_{i=1}^n{M}_{u_i,v_i}$ with $v_i{u}_j=0$ for $i\ge j$. Moreover, we give a complete characterization of locally quasi-nilpotent elementary operators of length 3.

15p