## 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} +σ_{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 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 language | English |
---|---|

Pages (from-to) | 613-632 |

Number of pages | 20 |

Journal | Journal of the London Mathematical Society |

Volume | 95 |

Issue number | 2 |

DOIs | |

State | Published - 1 Apr 2017 |

Externally published | Yes |

## Keywords

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

## ASJC Scopus subject areas

- General Mathematics