Certain Local Subsemigroups of Semigroups of Linear Transformations

A local subsemigroup of a semigroup $S$ is a subsemigroup of $S$ of the form $eAe$ where $A$ is a subsemigroup of $S$ and $e$ is an idempotent of $S$. It has been shown that for a finite nonempty set $X$ and an idempotent $\alpha$ of $T(X)$, $\alpha G(X)\alpha$ is a local subsemigroup of $T(X)$ if and only if either $\alpha$ is the identity mapping on $X$ or for every $a \in$ ran $\alpha, |a\alpha^{-1}| \ge |$ ran $\alpha|$ where $T(X)$ and $G(X)$ are the full transformation semigroup and the symmetric group on $X,$ respectively. In this paper, a parallel result is provided on the semigroup $L(V),$ under composition, of all linear transformations of a vector space $V.$ We show that for a finite- dimensional vector space $V$ and an idempotent $\alpha$ of $L(V), \alpha GL(V) \alpha$ is a local subsemigroup of $L(V)$ if and only if either $\alpha$ is the identity mapping on $V$ or dim$($ker $\alpha) \ge$ dim$($ran $\alpha)$ where $GL(V)$ is the group of isomorphisms of $V.$