Transitive subspaces (V. Lomonosov)

A set S of operators acts transitively on a vector space V if for all pairs of vectors x, y ∈ V with x ≠ 0 there is an element A ∈ S such that Ax = y. For a given positive integer k the set S acts k-transitively on V if for any linearly independent k vectors {x₁, x₂,..., x_k} and any k vectors {y₁, y₂,..., y_k} of V there is A ∈ S such that Ax_i = y_i, i = 1,..., k.
A classical theorem due to Burnside asserts that the only subalgebra of M_n(C) which acts transitively on Cⁿ is the matrix algebra itself. The situation is different for subspaces. For every 0 ≤ k < min{m, n}, there are k-transitive subspaces of M_mn(C) which are not (k+1)-transitive. We are going to discuss transitivity questions for subspaces of linear operators with different structures.