Non-commutative positive kernels and their matrix evaluations

We show that a formal power series in $2N$ non-commuting indeterminates is a positive non-commutative kernel if and only if the kernel on $N$-tuples of matrices of any size obtained from this series by matrix substitution is positive. We present two versions of this result related to different classes of matrix substitutions. In the general case we consider substitutions of jointly nilpotent $N$-tuples of matrices, and thus the question of convergence does not arise. In the ``convergent'' case we consider substitutions of $N$-tuples of matrices from a neighborhood of zero where the series converges. Moreover, in the first case the result can be improved: the positivity of a non-commutative kernel is guaranteed by the positivity of its values on the diagonal, i.e., on pairs of coinciding jointly nilpotent $N$-tuples of matrices. In particular this yields an analogue of a recent result of Helton on non-commutative sums-of-squares representations for the class of hereditary non-commutative polynomials. We show by an example that the improved formulation does not apply in the ``convergent'' case.