Journal article
Regular and positive noncommutative rational functions
Journal of the London Mathematical Society, v 95(2), pp 613-632
Apr 2017
Abstract
Call a noncommutative (nc) rational function r regular if it has no singularities, that is, r(X) is defined for all tuples of self-adjoint matrices X. In this paper, regular nc rational functions r are characterized via the properties of their (minimal size) linear systems realizations r = b* L-1 c. It is shown that r is regular if and only if L = A(0) + Sigma(j) A(j)x(j) is free elliptic. Roughly speaking, a linear pencil L is free elliptic if, after a finite sequence of basis changes and restrictions, the real part of A(0) is positive definite and the other A(j) are skew-adjoint. The second main result is a solution to an nc version of Hilbert's 17th problem: a positive regular nc rational function is a sum of squares.
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Details
- Title
- Regular and positive noncommutative rational functions
- Creators
- Igor Klep - University of AucklandJames Eldred Pascoe - Washington University in St. LouisJurij Volcic - University of Auckland
- Publication Details
- Journal of the London Mathematical Society, v 95(2), pp 613-632
- Publisher
- Wiley
- Number of pages
- 20
- Grant note
- University of Auckland DMS-1606260 / NSF; National Science Foundation (NSF) P1-0222; L1-6722 / Slovenian Research Agency; Slovenian Research Agency - Slovenia Marsden Fund Council of the Royal Society of New Zealand; Royal Society of New Zealand; Marsden Fund (NZ)
- Resource Type
- Journal article
- Language
- English
- Academic Unit
- Mathematics
- Web of Science ID
- WOS:000407967500013
- Scopus ID
- 2-s2.0-85020934053
- Other Identifier
- 991021861882204721
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- Collaboration types
- Domestic collaboration
- International collaboration
- Web of Science research areas
- Mathematics