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Non-commutative positive kernels and their matrix evaluations

Authors: Dmitry S. Kalyuzhnyi-Verbovetzkii and Victor Vinnikov
Journal: Proc. Amer. Math. Soc. 134 (2006), 805-816
MSC (2000): Primary 30C45, 47A56; Secondary 13F25, 47A60
Published electronically: July 19, 2005
MathSciNet review: 2180898
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Abstract: 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.

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Additional Information

Dmitry S. Kalyuzhnyi-Verbovetzkii
Affiliation: Department of Mathematics, Ben-Gurion University of the Negev, Beer Sheva, Israel 84105

Victor Vinnikov
Affiliation: Department of Mathematics, Ben-Gurion University of the Negev, Beer Sheva, Israel 84105

Keywords: Formal power series, non-commuting indeterminates, positive non-commutative kernels, matrix substitutions, hereditary polynomials, factorization
Received by editor(s): June 13, 2004
Received by editor(s) in revised form: October 19, 2004
Published electronically: July 19, 2005
Additional Notes: The first author was supported by the Center for Advanced Studies in Mathematics, Ben-Gurion University of the Negev.
The second author was partially supported by the Israel Science Foundation Grant 322/00-1
Communicated by: Joseph A. Ball
Article copyright: © Copyright 2005 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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