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.