Noncommutative positive kernels and their matrix evaluations
Authors:
Dmitry S. KalyuzhnyiVerbovetzkii and Victor Vinnikov
Journal:
Proc. Amer. Math. Soc. 134 (2006), 805816
MSC (2000):
Primary 30C45, 47A56; Secondary 13F25, 47A60
Published electronically:
July 19, 2005
MathSciNet review:
2180898
Fulltext PDF Free Access
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Abstract: We show that a formal power series in noncommuting indeterminates is a positive noncommutative 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 noncommutative 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 noncommutative sumsofsquares representations for the class of hereditary noncommutative 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. KalyuzhnyiVerbovetzkii
Affiliation:
Department of Mathematics, BenGurion University of the Negev, Beer Sheva, Israel 84105
Email:
dmitryk@math.bgu.ac.il
Victor Vinnikov
Affiliation:
Department of Mathematics, BenGurion University of the Negev, Beer Sheva, Israel 84105
Email:
vinnikov@math.bgu.ac.il
DOI:
http://dx.doi.org/10.1090/S000299390508127X
PII:
S 00029939(05)08127X
Keywords:
Formal power series,
noncommuting indeterminates,
positive noncommutative 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, BenGurion University of the Negev.
The second author was partially supported by the Israel Science Foundation Grant 322/001
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.
