ALL libraries (COBIB.SI union bibliographic/catalogue database)
PDF
  • Regular and positive noncommutative rational functions
    Klep, Igor, matematik ; Pascoe, James Eldred ; Volčič, Jurij, 1991-
    Call a noncommutative (nc) rational function ▫$\mathbbm{r}$▫ regular if it has no singularities, that is, ▫$\mathbbm{r}(X)$▫ is defined for all tuples of self-adjoint matrices ▫$X$▫. In this paper, ... regular nc rational functions ▫$\mathbbm{r}$▫ are characterized via the properties of their (minimal size) linear systems realizations ▫$\mathbbm{r} = \mathbf{b}^\ast L^{-1}\mathbf{c}$▫. It is shown that ▫$\mathbbm{r}$▫ is regular if and only if ▫$L = A_0 + \sum_j A_jx_j$▫ 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.
    Source: Journal of the London Mathematical Society. - ISSN 0024-6107 (Vol. 95, iss. 2, 2017, str. 613-632)
    Type of material - article, component part ; adult, serious
    Publish date - 2017
    Language - english
    COBISS.SI-ID - 18109017

    Link(s):

    http://dx.doi.org/10.1112/jlms.12030
    DOI

Shelf entry