admissibility

Let $k$ be a field, $V$ a vector space over $k$, and $T:V\to V$ a linear operator. We say that a subspace $W$ of $V$ is $T$-admissible if

1. $W$ is a $T$ - invariant subspace;

2. If $f\in k[X]$ (See the polynomial ring definition) and $f(T)x\in W$, there is a vector $y\in W$ such that $f(T)x=f(T)y$.