Positivstellensatz and flat functionals on path ∗-algebras

Popovych, Stanislav

doi:10.1016/j.jalgebra.2010.08.001

Cited by 6 publications

(4 citation statements)

References 11 publications

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“…A treatment of free noncommutative Hankel matrices is also presented in [Pop10]. There the existence of flat extensions, with necessary hypothesis, of noncommutative Hankel matrices which are merely positive semidefinite, rather than positive definite is established.…”

Section: 4mentioning

confidence: 99%

The convex Positivstellensatz in a free algebra

Helton

Klep

McCullough

2012

Advances in Mathematics

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Section: 4mentioning

confidence: 99%

The convex Positivstellensatz in a free algebra

Helton

Klep

McCullough

2012

Advances in Mathematics

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“…Free sets, matrix convex sets, linear pencils and LOI sets. This work fits into the wider context of free analysis [53,54,34,42,47,1,8,18,28,46], so we start by recalling some of the standard notions used throughout this article.…”

Section: Introductionmentioning

confidence: 99%

Operator Positivstellensätze for noncommutative polynomials positive on matrix convex sets

Zalar

2017

Journal of Mathematical Analysis and Applications

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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.

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“…We shall apply Proposition 3.1 in the next subsection to "flat" linear functionals, in which case the obtained quotient space H is finite-dimensional, and X is thus simply a tuple of matrices. We refer to [Pop10,HKM12] for more on flat linear functionals in a free algebra.…”

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confidence: 99%

Noncommutative polynomials nonnegative on a variety intersect a convex set

Helton¹,

Klep²,

Nelson³

2014

Journal of Functional Analysis

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Abstract. By a result of Helton and McCullough [HM12], open bounded convex free semialgebraic sets are exactly open (matricial) solution sets D • L of a linear matrix inequality (LMI) L(X) ≻ 0. This paper gives a precise algebraic certificate for a polynomial being nonnegative on a convex semialgebraic set intersect a variety, a so-called "Perfect" Positivstellensatz.For example, given a generic convex free semialgebraic set D • L we determine all "(strong sense) defining polynomials" p for D • L . Such polynomials must have the formwhere q i , r j are matrices of polynomials, and C j are real matrices satisfying C j L = LC j . This follows from our general result for a given linear pencil L and a finite set I of rows of polynomials. A matrix polynomial p is positive where L is positive and all ι ∈ I vanish if and only ifwhere each p i , q j and r k are matrices of polynomials of appropriate dimension, and each ι k is an element of the "L-real radical" of I. In this representation, we can restrict p i , q i , ι k and r k to be elements of a low-dimensional subspace of matrices of polynomials, and in particular, their degrees depend in a very tame way only on the degree of p and the degrees of the elements of I. Further, this paper gives an efficient algorithm for computing the L-real radical of I. This Positivstellensatz extends and unifies two different lines of results: (1) the free real Nullstellensatz of [CHMN13, Nel] which gives an algebraic certificate corresponding to one polynomial being zero on the free variety where others are zero; this is (P) with L = 1; (2) the convex Positivstellensatz of [HKM12, KS11] which is (P) without I; i.e., I = {0}. The representation (P) has a number of additional consequences which will be presented.

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Positivstellensatz and flat functionals on path ∗-algebras

Cited by 6 publications

References 11 publications

The convex Positivstellensatz in a free algebra

The convex Positivstellensatz in a free algebra

Operator Positivstellensätze for noncommutative polynomials positive on matrix convex sets

Noncommutative polynomials nonnegative on a variety intersect a convex set

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