On hermitian polynomial optimization

Putinar, Mihai

doi:10.1007/s00013-006-1641-x

Cited by 5 publications

(2 citation statements)

References 20 publications

(26 reference statements)

Supporting

Mentioning

Contrasting

Order By: Relevance

“…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

Self Cite

View full text Add to dashboard Cite

show abstract

Section: Commutative Hereditary Sums Of Squaresmentioning

confidence: 99%

Strong majorization in a free ✱-algebra

2006

Self Cite

View full text Add to dashboard Cite

show abstract

“…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

View full text Add to dashboard Cite

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.

show abstract

Polynomial Optimization on Odd-Dimensional Spheres

D’Angelo

Putinar

2008

Emerging Applications of Algebraic Geometry

Self Cite

View full text Add to dashboard Cite

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.

show abstract

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

Contact Info

Product

Resources

About