Operator Positivstellensatze for noncommutative polynomials positive on matrix convex sets

JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS

卷:445

Article

Zalar, Aljaz¹ 

[1] Inst Math Phys & Mech, Jadranska 19, Ljubljana 1000, Slovenia

关键词: Free convexity;    Linear matrix inequality (LMI);    Spectrahedron;    Completely positive;    Positivstellensatz;    Free real algebraic geometry;   

DOI  :  10.1016/j.jmaa.2016.07.043

来源: Elsevier

【 摘 要 】

This article studies algebraic certificates of positivity for noncommutative (nc) operator-valued polynomials on matrix convex sets, such as the solution set D-L, called a free Hilbert spectrahedron, of the linear operator inequality (LOI) L(X) = A(0) circle times I+Sigma(g)(j=1) Aj circle times X-j greater than or similar to 0, where A(j) are self-adjoint linear operators on a, separable Hilbert space, X-j matrices and I is an identity matrix. If A(j) are matrices, then L(X) greater than or similar to 0 is called a linear matrix inequality (LMI) and D-L a free spectrahedron. For monic LMIs, i.e., A(0) = I, and nc matrix-valued polynomials the certificates of positivity were established by Helton, Klep and McCullough in a series of articles with the use of the theory of complete positivity from operator algebras and classical separation arguments from real algebraic geometry. Since the full strength of the theory of complete positivity is not restricted to finite dimensions, but works well also in the infinite-dimensional setting, we use it to tackle our problems. First we extend the characterization of the inclusion D-L1 subset of D-L2 from monic LMIs to monic LOIs L-1 and L-2. As a corollary one immediately obtains the description of a polar dual of a free Hilbert spectrahedron D-L, and its projection, called a free Hilbert spectrahedrop. Further on, using this characterization in a separation argument, we obtain a certificate for multivariate matrix-valued nc polynomials F positive semidefinite on a free Hilbert spectrahedron defined by a monic LOI. Replacing the separation argument by an operator Fejer-Riesz theorem enables us to extend this certificate, in the univariate racy, to operator-valued polynomials F. Finally, focusing on the algebraic description of the equality D-L1 = D-L2, we remove the assumption of boundedness from the description in the LMIs case by an extended analysis. However, the description does not extend to LOIs case by counterexamples. (C) 2016 Elsevier Inc. All rights reserved.

【 授权许可】

Free