Semigroups of partial linear transformations whose restrictions belong to an injective partial linear transformation semigroup

Let $V$ be a vector space over a field $\mathbb{F}$ and let $W$ be a subspace of $V$. The semigroup of partial linear transformations on $V$ whose restriction to $W$ belongs to an injective partial linear transformation semigroup $\mathcal{I}(W)$ is denoted by $P_{\mathcal{I}(W)}(V)$. In this paper, we describe Green's relations for $P_{\mathcal{I}(W)}(V)$, characterize its regular elements, and give necessary and sufficient conditions for $P_{\mathcal{I}(W)}(V)$ to be regular, inverse, or completely regular. We also analyze the ideal structure of $P_{\mathcal{I}(W)}(V)$, identifying its maximal and minimal ideals.