Let V be a vector space of dimension N over the finite field Fq and T be a linear operator on V . Given an integer m that divides N, an m-dimensional subspace W of V is T -splitting if V = W ⊕ TW ⊕ · · · ⊕ T^(d−1) W where d = N/m. Let σ(m, d; T ) denote the number of m-dimensional T -splitting subspaces. Determining σ(m, d; T ) for an arbitrary operator T is an open problem.
We prove that σ(m, d; T ) depends only on the similarity class type of T and give an explicit formula in the special case where T is cyclic and nilpotent. Denote by σ_q(m, d; τ) the number of m-dimensional splitting subspaces for a linear operator of similarity class type τ over an Fq-vector space of dimension md. For fixed values of m, d and τ, we show that σ_q(m, d; τ) is a polynomial in q.