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

DOI:
https://doi.org/10.1090/S0002-9939-05-08127-X

Published electronically:
July 19, 2005

MathSciNet review:
2180898

Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: 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.

**1.**D. Alpay, A. Dijksma, J. Rovnyak, and H. de Snoo,*Schur functions, operator colligations, and reproducing kernel Pontryagin spaces*, Operator Theory: Adv. Appl. 96, Birkhäuser Verlag, Basel, 1997. MR**1465432 (2000a:47024)****2.**D. Alpay and D.S. Kalyuzhnyi-Verbovetzkii,*On the intersection of null spaces for matrix substitutions in a non-commutative rational formal power series*, C. R. Math. Acad. Sci. Paris**339**(2004), no. 8, 533-538.**3.**N. Aronszajn,*Theory of reproducing kernels*, Trans. Amer. Math. Soc.**68**(1950), 337-404. MR**0051437 (14:479c)****4.**J.A. Ball, G. Groenewald, and T. Malakorn,*Conservative structured noncommutative multidimensional linear systems*, Multidimens. Syst. Signal Process., to appear.**5.**J.A. Ball and V. Vinnikov,*Formal reproducing kernel Hilbert spaces: The commutative and noncommutative settings*, in*Reproducing kernel spaces and applications*, 77-134, Operator Theory Adv. Appl. 143, Birkhäuser Verlag, Basel, 2003. MR**2019348 (2005a:46058)****6.**J.W. Helton,*``Positive'' noncommutative polynomials are sums of squares*, Ann. of Math. (2)**156**(2002). MR**1933721 (2003k:12002)****7.**J.W. Helton, S. McCullough, and M. Putinar,*A non-commutative Positivstellensatz on isometries*, J. Reine Angew. Math.**568**(2004), 71-80. MR**2034923 (2004m:47026)****8.**S. McCullough,*Factorization of operator-valued polynomials in several non-commuting variables*, Linear Algebra Appl.**326**(2001), no. 1-3, 193-203. MR**1815959 (2002f:47035)****9.**S. Saitoh,*Theory of Reproducing Kernels and its Applications*, Pitman Res. Notes Math. 189, Longman Scientific and Technical, Harlow, 1988. MR**0983117 (90f:46045)****10.**B.V. Shabat,*Introduction to Complex Analysis. Part II*, Translations of Mathematical Monographs 110, Functions of several variables, Translated from the third (1985) Russian edition by J. S. Joel, American Mathematical Society, Providence, RI, 1992. MR**1192135 (93g:32001)**

Retrieve articles in *Proceedings of the American Mathematical Society*
with MSC (2000):
30C45,
47A56,
13F25,
47A60

Retrieve articles in all journals with MSC (2000): 30C45, 47A56, 13F25, 47A60

Additional Information

**Dmitry S. Kalyuzhnyi-Verbovetzkii**

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

Email:
dmitryk@math.bgu.ac.il

**Victor Vinnikov**

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

Email:
vinnikov@math.bgu.ac.il

DOI:
https://doi.org/10.1090/S0002-9939-05-08127-X

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.