Regular and positive noncommutative rational functions:

Igor Klep, James Eldred Pascoe, Jurij Volčič

Research output: Contribution to journalArticlepeer-review

11 Scopus citations


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-1c. It is shown that r is regular if and only if L = A0j Ajxj 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 A0 is positive definite and the other Aj 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.

Original languageEnglish
Pages (from-to)613-632
Number of pages20
JournalJournal of the London Mathematical Society
Issue number2
StatePublished - 1 Apr 2017
Externally publishedYes


  • 13J30
  • 15A22
  • 16K40
  • 26C15
  • 47A63 (secondary)
  • 47L07 (primary)

ASJC Scopus subject areas

  • General Mathematics


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