Generalized commutators and a problem related to the Amitsur–Levitzki theorem

The generalized commutator [ A1 vertical bar center dot center dot center dot vertical bar A(k)] of a list A1,..., A(k) of k real n x n matrices is defined as a multilinear skew-symmetric function and the linear operator T = T(A1,..., A(k)) on the vector space Mn(R) is defined by TX := [ A1 vertical bar center dot center dot center dot vertical bar A(k) vertical bar X]. The Amitsur-Levitzki theorem shows that T = 0 when k = 2n-1. We investigate the kernel of T and prove that for all integers k and n such that 2 = k = 2n-2 we have dim ker T(A1,..., A(k)) =.0(n, k) where.0(n, k) := k if k is even; k + 1 if k is odd and n is even; and k + 2 if k and n are both odd. We conjecture that this result is best possible and that dim ker T(A1,..., A(k))=.0(n, k) for almost all A1,..., A(k) when k and n are in this range. This conjecture is supported by some computational evidence but so far remains open.