Operator Positivstellensätze for noncommutative polynomials positive on matrix convex sets

Abstract

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 ⊗ I + g j=1 A j ⊗ X j 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) 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 L 1 ⊆ D L 2 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 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 D L 1 = D L 2 , 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.

Minimal and maximal matrix convex sets

“…. , X d , X d+1 ) = d j=1 A j ⊗ X j + I ⊗ X d+1 , and its corresponding positivity set Dh L A = {X ∈ M d+1 sa : h L A (X) ≥ 0} (see [24]). The matrix range [3, Section 2.4] of a tuple A in B(H) d is defined to be the set Minimal and maximal matrix convex sets.…”

Shape, scale, and minimality of matrix ranges

“…We note that if H is assumed finite-dimensional, then more general uniqueness results were proved in [3,16,24], either in terms of matrix ranges or free spectrahedra. Therefore, Corollary 3.13 is primarily of use to determine the shape of K based on T (or vice-versa) when H is infinite-dimensional, or to determine when the dimension of H must be finite or infinite based on other assumptions.…”

Bianalytic free maps between spectrahedra and spectraballs

“…. . , Ξ g ) ∈ M g (C) g such that (1.1) Given A ∈ M r (C) g , we say L A (or L re A ) is minimal for a free spectrahedron D if D = D A and if for any other B ∈ M r ′ (C) g satisfying D = D B it follows that r ′ ≥ r. A minimal L A for D A exists and is unique up to unitary equivalence [HKM13,Zal17]. We can now state Theorem 1.1, our principal result on bianalytic mappings from a spectraball onto a free spectrahedron.…”