On hermitian polynomial optimization

Putinar, Mihai

doi:10.1007/s00013-006-1641-x

Cited by 5 publications

(2 citation statements)

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“…If we put by convention all adjoints X * j to the left of X k , in q(X, X * ), then the same proof applies and yields the following result. For more details about such (hermitian, or subharmonic) decompositions, see [12]. For noncommutative hereditary polynomials the analogous theorem is used in Nick Slinglend's UCSD thesis in studying the effect of noncommutative biholomorphic maps.…”

Section: Commutative Hereditary Sums Of Squaresmentioning

confidence: 99%

Strong majorization in a free ✱-algebra

2006

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Section: Commutative Hereditary Sums Of Squaresmentioning

confidence: 99%

Strong majorization in a free ✱-algebra

2006

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“…In the present paper, we show that finite-dimensional contractive realizations of a rational matrix-valued function F exist when F is regular on the closed domain D P and the Agler norm F A,P is strictly less than 1 if P = k i=1 P i and the matrix polynomials P i satisfy a certain natural Archimedean condition. The proof has two ingredients: a matrix-valued version of a Hermitian Positivstellensatz [18] (see also [13,Corollary 4.4]), and a lurking contraction argument. For the first ingredient, we introduce the notion of a matrix system of Hermitian quadratic modules and the Archimedean property for them, and use the hereditary functional calculus for evaluations of a Hermitian symmetric matrix polynomial on d-tuples of commuting operators on a Hilbert space.…”

Section: Introductionmentioning

confidence: 99%

Matrix-valued Hermitian Positivstellensatz, lurking contractions, and contractive determinantal representations of stable polynomials

Grinshpan¹,

Kaliuzhnyi-Verbovetskyi²,

Vinnikov³

et al. 2015

Preprint

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We prove that every matrix-valued rational function F , which is regular on the closure of a bounded domain D P in C d and which has the associated Agler norm strictly less than 1, admits a finite-dimensional contractive realizationHere D P is defined by the inequality P(z) < 1, where P(z) is a direct sum of matrix polynomials P i (z) (so that appropriate Archimedean and approximation conditions are satisfied), and P(z) n = k i=1 P i (z) ⊗ I ni , with some k-tuple n of multiplicities n i ; special cases include the open unit polydisk and the classical Cartan domains. The proof uses a matrix-valued version of a Hermitian Positivstellensatz by Putinar, and a lurking contraction argument. As a consequence, we show that every polynomial with no zeros on the closure of D P is a factor of det(I − KP(z) n ), with a contractive matrix K.

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Polynomial Optimization on Odd-Dimensional Spheres

D’Angelo

Putinar

2008

Emerging Applications of Algebraic Geometry

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Abstract. The sphere S 2d−1 naturally embeds into the complex affine spaceWe show how the complex variables in C d simplify the known Striktpositivstellensätze, when the supports are resticted to semi-algebraic subsets of odd dimensional spheres. Mathematics Subject Classification (2000). Primary 14P10; Secondary 32A70.Keywords. Positive polynomial, Hermitian square, unit sphere, plurisubharmonic function. PreliminariesLet C d denote complex Euclidean space with Euclidean norm given by |z| 2 = d j=1 |z j | 2 . The unit, odd dimensional sphereis a particularly important example of a Cauchy-Riemann (usually abbreviated CR) manifold. This note will show how one can study problems of polynomial optimization over semi-algebraic subsets of S 2d−1 by using the induced CauchyRiemann structure. Our results can be regarded as multivariate analogues of classical phenomena about positive trigonometric polynomials, known for a long time in dimension one (d = 1). They are also related to results concerning proper holomorphic mappings between balls in different dimensional complex Euclidean spaces and the geometry of holomorphic vector bundles.

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On hermitian polynomial optimization

Cited by 5 publications

References 20 publications

Strong majorization in a free ✱-algebra

Strong majorization in a free ✱-algebra

Matrix-valued Hermitian Positivstellensatz, lurking contractions, and contractive determinantal representations of stable polynomials

Polynomial Optimization on Odd-Dimensional Spheres

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