116 citations · 1.1k across the 72 of their papers we have counts for
26 papers · 1 filter
Addendum to "Factoring non-negative operator valued trigonometric polynomials in two variables"
Michael A. Dritschel, Igor Klep, Scott McCullough +1
Factorization for positive semidefinite matrix-valued polynomials over a nonsingular compact affine real surface is established. Corollaries include Fejér-Riesz factorization for b…
Operator-Valued Positivstellensätze on Matrix Convex Sets and Free Products of Finite Abelian Groups
Abhay Jindal, Igor Klep, Scott McCullough
We prove a Positivstellensatz for operator-valued noncommutative polynomials that are positive on matrix convex sets. Specifically, let be an operator-valued polynomial in $B(H…
Fejér--Riesz factorization for positive noncommutative trigonometric polynomials
Igor Klep, Jacob Levenson, Scott McCullough
We prove a Fejér-Riesz type factorization for positive matrix-valued noncommutative trigonometric polynomials on , where is either the…
Positive operator-valued noncommutative polynomials are squares
Abhay Jindal, Igor Klep, Scott McCullough
We establish operator-valued versions of the earlier foundational factorization results for noncommutative polynomials due to Helton (Ann.~Math., 2002) and one of the authors (Line…
Cross-positive linear maps, positive polynomials and sums of squares
Igor Klep, Klemen Šivic, Aljaž Zalar
A linear map between matrix spaces is called cross-positive if it is positive on orthogonal pairs of positive semidefinite matrices in the sense that $\langle U,V\rangl…
Sums of squares certificates for polynomial moment inequalities
Igor Klep, Victor Magron, Jurij Volčič
This paper introduces and develops the algebraic framework of moment polynomials, which are polynomial expressions in commuting variables and their formal mixed moments. Their posi…
A random copositive matrix is completely positive with positive probability
Igor Klep, Tea Štrekelj, Aljaž Zalar
An symmetric matrix is copositive if the quadratic form is nonnegative on the nonnegative orthant. The cone of copositive matrices strictly contains the con…
State polynomials: positivity, optimization and nonlinear Bell inequalities
Igor Klep, Victor Magron, Jurij Volčič +1
This paper introduces state polynomials, i.e., polynomials in noncommuting variables and formal states of their products. A state analog of Artin's solution to Hilbert's 17th probl…
Matrix Extreme Points and Free extreme points of Free spectrahedra
Aidan Epperly, Eric Evert, J. William Helton +1
A spectrahedron is a convex set defined by a linear matrix inequality, i.e., the set of all such that \[ L_A(x) = I + A_1 x_1 + A_2 x_2 + \dots + A_g x_g \succ…
Facial structure of matrix convex sets
Igor Klep, Tea Štrekelj
This article investigates the notions of exposed points and (exposed) faces in the matrix convex setting. Matrix exposed points in finite dimensions were first defined by Kriel in…
Noncommutative partially convex rational functions
Michael Jury, Igor Klep, Mark E. Mancuso +2
Motivated by classical notions of bilinear matrix inequalities (BMIs) and partial convexity, this article investigates partial convexity for noncommutative functions. It is shown t…
Noncommutative partial convexity via -convexity
Michael Jury, Igor Klep, Mark E. Mancuso +2
Motivated by classical notions of partial convexity, biconvexity, and bilinear matrix inequalities, we investigate the theory of free sets that are defined by (low degree) noncommu…
Plurisubharmonic Noncommutative Rational Functions
Harry Dym, J. William Helton, Igor Klep +2
A noncommutative (nc) function in is called plurisubharmonic (plush) if its nc complex Hessian takes only positive semidefinite values on an nc neig…
Noncommutative polynomials describing convex sets
J. W. Helton, I. Klep, S. McCullough +1
The free closed semialgebraic set determined by a hermitian noncommutative polynomial is the closure of the connected component of containing…
Bianalytic free maps between spectrahedra and spectraballs
J. William Helton, Igor Klep, Scott McCullough +1
Linear matrix inequalities (LMIs) are ubiquitous in real algebraic geometry, semidefinite programming, control theory and signal processing. LMIs with (dimension free) matrix unkno…
Free bianalytic maps between spectrahedra and spectraballs in a generic setting
Meric Augat, J. William Helton, Igor Klep +1
Given a tuple of matrices, the collection of those tuples of matrices (of the same size) such that $\| \sum E_j\otimes X_j\|\le…
There are many more positive maps than completely positive maps
Igor Klep, Scott McCullough, Klemen Šivic +1
A linear map between matrix spaces is positive if it maps positive semidefinite matrices to positive semidefinite ones, and is called completely positive if all its ampliations are…
Bianalytic Maps Between Free Spectrahedra
Meric Augat, J. William Helton, Igor Klep +1
Linear matrix inequalities (LMIs) play a role in many areas of applications and the set of solutions to one is called…
Dilations, Linear Matrix Inequalities, the Matrix Cube Problem and Beta Distributions
J. William Helton, Igor Klep, Scott A. McCullough +1
An operator C on a Hilbert space H dilates to an operator T on a Hilbert space K if there is an isometry V from H to K such that C=V^*TV. A main result of this paper is, for a posi…
Matrix Convex Hulls of Free Semialgebraic Sets
J. William Helton, Igor Klep, Scott McCullough
This article resides in the realm of the noncommutative (free) analog of real algebraic geometry - the study of polynomial inequalities and equations over the real numbers - with a…
Noncommutative polynomials nonnegative on a variety intersect a convex set
J. William Helton, Igor Klep, Christopher S. Nelson
By a result of Helton and McCullough, open bounded convex free semialgebraic sets are exactly open (matricial) solution sets D_L of a linear matrix inequality (LMI) L(X)>0. This pa…
Free Convex Algebraic Geometry
J. William Helton, Igor Klep, Scott McCullough
This chapter is a tutorial on techniques and results in free convex algebraic geometry and free real algebraic geometry (RAG). The term free refers to the central role played by al…
Correction of a proof in "Connes' embedding conjecture and sums of hermitian squares"
Sabine Burgdorf, Ken Dykema, Igor Klep +1
We show that Connes' embedding conjecture (CEC) is equivalent to a real version of the same (RCEC). Moreover, we show that RCEC is equivalent to a real, purely algebraic statement…
Free analysis, convexity and LMI domains
J. William Helton, Igor Klep, Scott McCullough
This paper concerns free analytic maps on noncommutative domains. These maps are free analogs of classical holomorphic functions in several complex variables, and are defined in te…