Is every subspace can be represented as T-cyclic for some T?

Question

Definition of $T$-cyclic subspace: Let $T$ be an linear operator on $V$. Take $v \in V$.

The subspace generated by $\text{span}(\{v,T(v),T^2(v),\cdots\})$ is called $T$-cyclic subspace generated by $v$.

My question is: for a subspace $W$ on $V$, does there exists a linear operator $T$, such that $W$ is $T$-cyclic subspace for some $v \in W$?.

If my question is foolish one. Sorry in advance.