Contributions to a noncommutative real algebraic geometry : doctoral thesis

english

doktorska dela

2017

Univerza v Ljubljani, Fakulteta za matematiko in fiziko
(Obvezni izvod spletne publikacije)

We study Positivstellensätze from noncommutative real algebraic geometry. Of these, we focus on two specific ones. A version of the matrix Fejér-Riesz theorem characterizes positive semidefinite matrix polynomials on the real line. This characterization has already been extended from the real line to a disjoint union of finitely many closed intervals in the case of scalar polynomials and to a single closed interval in the case of matrix polynomials. Our first interest in this thesis is to figure out what can be said in the case of matrix polynomials and a disjoint union of finitely many closed intervals. Algebraic certificates of positivity for noncommutative matrix polynomials on matrix convex sets, such as the solution set of a linear matrix inequality (LMI), have recently attracted much attention among real algebraic geometers. In the case of LMIs many certificates are known. Since every closed matrix convex set containing the origin is the solution set of a linear operator inequality (LOI), this attracts the second interest of this thesis which is to extend the certificates from matrix to operator polynomials. Our main result referring to the first problem is a denominator-free characterization in the case of a compact union, called a Compact Positivstellensatz. The technique in the proof is the adaptation of Schur complements and eliminating the denominators with the help of known results for scalar polynomials. We also construct counterexamples for the extension of the characterization to almost all non-compact unions. By developing the connections between matrix polynomials and Laurent matrix polynomials we obtain the matrix Positivstellensatz on a disjoint union of finitely many closed arcs in the unit complex circle and finally, using this result we come to a Non-compact Positivstellensatz for a non-compact union of finitely many closed intervals in the real line using only simple denominators. Referring to the second problem our first result is an algebraic characterization for the domination of the solution sets of monic LOIs, called a Linear Positivstellensatz. The techniques used are complete positivity and the theory of operator algebras. We provide examples which show that the monicity assumption is necessary. As a consequence we also obtain the description of the polar dual of the LOI. Next we focus on the question of the equality of the solution sets of two LOIs which turns out to be a harder one. We present the answer for LOIs with compact operator coeficients, called a Linear Gleichstellensatz. Namely, under some minimality assumption, the LOIs are unitarily equivalent. The idea is to understand the unital ?$C^\ast$?-algebras generated by the coefficients and ?$\ast$?-isomorphisms between them. We show by examples that the answer does not extend to arbitrary LOIs. Finally, we establish a Convex Positivstellensatz which characterizes matrix polynomials positive semidefinite on the solution set of a LOI and show that in the univariate case it extends to operator polynomials.

12.05.2023