Unit and idempotent additive maps over countable linear transformations

Abstract

Let $V$ be a countably generated right vector space over a field $F$ and $\sigma\in End(V_F )$ be a shift operator. We show that there exist a unit $u$ and an idempotent $e$ such that $1-u,\sigma-u$ are units in $End(V_F)$ and $1-e,\sigma-e$ are idempotents in $End(V_F)$. We also obtain that if $D$ is a division ring and $D\ncong \mathbb Z_2, \mathbb Z_3 $, then $1-u,\alpha-u$ are units in $End(V_D)$ for any $\alpha\in End(V_D)$.