## Non-commutative positive kernels and their matrix evaluations

HTML articles powered by AMS MathViewer

- by Dmitry S. Kalyuzhnyĭ-Verbovetzkiĭ and Victor Vinnikov PDF
- Proc. Amer. Math. Soc.
**134**(2006), 805-816 Request permission

## 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.## References

- Daniel Alpay, Aad Dijksma, James Rovnyak, and Hendrik de Snoo,
*Schur functions, operator colligations, and reproducing kernel Pontryagin spaces*, Operator Theory: Advances and Applications, vol. 96, Birkhäuser Verlag, Basel, 1997. MR**1465432**, DOI 10.1007/978-3-0348-8908-7 - D. Alpay and D.S. Kalyuzhnyĭ-Verbovetzkiĭ,
*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. - N. Aronszajn,
*Theory of reproducing kernels*, Trans. Amer. Math. Soc.**68**(1950), 337–404. MR**51437**, DOI 10.1090/S0002-9947-1950-0051437-7 - J.A. Ball, G. Groenewald, and T. Malakorn,
*Conservative structured noncommutative multidimensional linear systems*, Multidimens. Syst. Signal Process., to appear. - Joseph A. Ball and Victor Vinnikov,
*Formal reproducing kernel Hilbert spaces: the commutative and noncommutative settings*, Reproducing kernel spaces and applications, Oper. Theory Adv. Appl., vol. 143, Birkhäuser, Basel, 2003, pp. 77–134. MR**2019348** - J. William Helton,
*“Positive” noncommutative polynomials are sums of squares*, Ann. of Math. (2)**156**(2002), no. 2, 675–694. MR**1933721**, DOI 10.2307/3597203 - J. William Helton, Scott A. McCullough, and Mihai Putinar,
*A non-commutative Positivstellensatz on isometries*, J. Reine Angew. Math.**568**(2004), 71–80. MR**2034923**, DOI 10.1515/crll.2004.019 - Scott McCullough,
*Factorization of operator-valued polynomials in several non-commuting variables*, Linear Algebra Appl.**326**(2001), no. 1-3, 193–203. MR**1815959**, DOI 10.1016/S0024-3795(00)00285-8 - Saburou Saitoh,
*Theory of reproducing kernels and its applications*, Pitman Research Notes in Mathematics Series, vol. 189, Longman Scientific & Technical, Harlow; copublished in the United States with John Wiley & Sons, Inc., New York, 1988. MR**983117** - B. V. Shabat,
*Introduction to complex analysis. Part II*, Translations of Mathematical Monographs, vol. 110, American Mathematical Society, Providence, RI, 1992. Functions of several variables; Translated from the third (1985) Russian edition by J. S. Joel. MR**1192135**, DOI 10.1090/mmono/110

## Additional Information

**Dmitry S. Kalyuzhnyĭ-Verbovetzkiĭ**- 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
- MR Author ID: 224109
- Email: vinnikov@math.bgu.ac.il
- 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
- © Copyright 2005
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication. - 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
- MathSciNet review: 2180898