Conditions under which the order of two non-negative Hermitian matrices is preserved when taking integral powers

Let A and B be two n×n non-negative matrices. We write A ? B iff

$\text{u}{}^1\text{(A ? B)u ? 0}$

for all column vectors u in Cⁿ. Here u¹ is the conjugate transpose of u. In this paper are stated equivalent conditions under which A^k ? B^k for all natural numbers k. The result is then generalized to Hermitian operators in a Hilbert space.