  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 $d=n/m$. 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.
