Polynomial Matrices, Splitting Subspaces and Krylov Subspaces over Finite Fields

Let $T$ be a linear operator on an ${\mathbb{F}}_q$-vector space $V$ of dimension $n$. For any divisor $m$ of $n$, an $m$-dimensional subspace $W$ of $V$ is $T$-splitting if $$V=W\oplus TW\oplus \cdots \oplus T^{d-1}W,$$ where $d=n\mathrm{/}m$. Let $\sigma (m,d;T)$ denote the number of $m$-dimensional $T$-splitting subspaces. Determining $\sigma (m,d;T)$ for an arbitrary operator $T$ is an open problem. This problem is closely related to another open problem on Krylov spaces. We discuss this connection and give explicit formulae for $\sigma (m,d;T)$ in the case where the invariant factors of $T$ satisfy certain degree conditions. A connection with another enumeration problem on polynomial matrices is also discussed.