In Advanced Modern Algebra, Rotman gives the following construction of a module that he denotes by $V^T$: Let $V$ be a vector space over a field $k$ and let $T:V \rightarrow V$ be a linear operator on $V$. Then, for a given polynomial $f(x) = \sum_i c_ix^i \in k[x]$ and $v \in V$ define scalar multiplication according to $$ \cdot :k[x] \times V \rightarrow V : f \cdot v \mapsto \sum_i c_i T^i(v) $$ where $T^i$ denotes the $i$-fold composition of $T$ with itself. It is clear that $V$ together with the indicated scalar multiplication is a $k[x]$-module.
My question is the following: Is this construction important enough to have a name and, if so, what is it called and how does it fit into the bigger picture? What is the motivation for this construction?