Scott McCullough

A positivstellensatz for non-commutative polynomials

A non-commutative polynomial which is positive on a bounded semi-algebraic set of operators has a weighted sum of squares representation. This Positivstellensatz parallels similar results in the commutative case. A broader issue is, to what extent does real semi-algebraic geometry extend to non-commutative polynomials? Our strict Positivstellensatz is positive news, on the opposite extreme from strict positivity would be a Real Nullstellensatz. We give an example which shows that there is no non-commutative Real Nullstellensatz along certain lines. However, we include a successful type of non-commutative Nullstellensatz proved by George Bergman.

Factorization of operator-valued polynomials in several non-commuting variables☆

Abstract A version of Fejer–Riesz factorization and factorization of positive operator-valued polynomials in several non-commuting variables holds. The proofs use Arvesons extension theorem and matrix completions.

The matricial relaxation of a linear matrix inequality

AbstractGiven linear matrix inequalities (LMIs) L1 and L2 it is natural to ask: (Q1) when does one dominate the other, that is, does

The failure of rational dilation on a triply connected domain

{L_1(X) \succeq 0}

Engineering Systems and Free Semi-Algebraic Geometry

imply

Convex Matrix Inequalities Versus Linear Matrix Inequalities

{L_2(X) \succeq 0}