刊名:Journal of Mathematical Analysis and Applications
出版年:2017
出版时间:1 January 2017
年:2017
卷:445
期:1
页码:32-80
全文大小:800 K
文摘
This article studies algebraic certificates of positivity for noncommutative (nc) operator-valued polynomials on matrix convex sets, such as the solution set fccf5876ac7d451a0a2f7" title="Click to view the MathML source">DL, called a free Hilbert spectrahedron, of the linear operator inequality (LOI) fcced632a57fa37781f38368b9e">, where fc6a6bb00fc1a25" title="Click to view the MathML source">Aj are self-adjoint linear operators on a separable Hilbert space, Xj matrices and I is an identity matrix. If fc6a6bb00fc1a25" title="Click to view the MathML source">Aj are matrices, then L(X)⪰0 is called a linear matrix inequality (LMI) and fccf5876ac7d451a0a2f7" title="Click to view the MathML source">DL a free spectrahedron. For monic LMIs, i.e., A0=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 DL1⊆DL2 from monic LMIs to monic LOIs L1 and L2. As a corollary one immediately obtains the description of a polar dual of a free Hilbert spectrahedron fccf5876ac7d451a0a2f7" title="Click to view the MathML source">DL 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 Fejér–Riesz theorem enables us to extend this certificate, in the univariate case, to operator-valued polynomials F . Finally, focusing on the algebraic description of the equality DL1=DL2, 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.