Non-commutative positive kernels and their matrix evaluations

Details Non-commutative positive kernels and their matrix evaluations Non-commutative positive kernels and their matrix evaluations

2004-12-08

arxiv.org/abs/math/0412165v1

Mathematics

Functional Analysis

Scientific paper

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.

Affiliated with

Also associated with

No associations

Rating

Non-commutative positive kernels and their matrix evaluations does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.

If you have personal experience with Non-commutative positive kernels and their matrix evaluations, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and Non-commutative positive kernels and their matrix evaluations will most certainly appreciate the feedback.

Rate now

Comments { 0 }

Profile ID: LFWR-SCP-O-169999