Factoring non-negative matrix valued trigonometric polynomials in two variables

Analyse Fonctionnelle

Salle Kampé de Fériet M2
Michael Dritschel
Vendredi, 30 Mars, 2018 - 14:00 - 15:00
The Fejér-Riesz theorem states that a non-negative trigonometric polynomial
is the hermitian square of an analytic polynomial of the same degree. 
Rosenblum showed that the theorem is still valid if the coefficents are
Hilbert space operators. The speaker later extended this to strictly
positive operator valued trigonometric polynomials in finitely many
variables. Results in real algebra due to Scheiderer imply that scalar
valued non-negative trigonometric polynomials in two variables always factor
as a finite sum of squares of analytic polynomials, and that this fails in
three or more variables.  We discuss a purely analytic approach, using Schur
complement techniques, to showing that any non-negative matrix valued
trigonometric polynomial in two variables is a finite sum of squares of
analytic polynomials. In analogy with the Tarski transfer principle in real
algebra, the proof lifts the problem to an ultraproduct, solves it there,
and then shows that this implies the existence of a solution in the original
context.  While the method is non-constructive, it nevertheless implies a
concrete algorithm for such a factorization.