Paratransitive algebras of linear operators

In this article we study a natural weakening – which we refer to as paratransitivity – of the well-known notion of transitivity of an algebra AA of linear operators acting on a finite-dimensional vector space VV. Given positive integers k and m, we shall say that such an algebra AA is (k,m)(k,m)-transitive if for every pair of subspaces W1W1 and W2W2 of VV of dimensions k and m respectively, we have AW1∩W2≠{0}AW1∩W2≠{0}. We consider the structure of minimal (k,m)(k,m)-transitive algebras and explore the connection of this notion to a measure of largeness for invariant subspaces of A.