November 8, 2004
Dmitry Kalyuzhnyi-Verbovetzkii
Title: Non-commutative positive kernels and their matrix evaluations
This talk is based on a joint work with Victor Vinnikov.
We show that a formal power series in 
non-commuting indeterminates
is a positive non-commutative kernel if and only if
the kernel on 
-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 -tuples of matrices, and thus the question
of convergence does not arise. In the ``convergent'' case we consider
substitutions of -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 -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.