TY - JOUR

T1 - Matrix-valued hermitian positivstellensatz, lurking contractions, and contractive determinantal representations of stable polynomials

AU - Grinshpan, Anatolii

AU - Kaliuzhnyi-Verbovetskyi, Dmitry S.

AU - Vinnikov, Victor

AU - Woerdeman, Hugo J.

N1 - Publisher Copyright:
© 2016 Springer International Publishing.

PY - 2016/1/1

Y1 - 2016/1/1

N2 - We prove that every matrix-valued rational function F, which is regular on the closure of a bounded domain DP in Cd and which has the associated Agler norm strictly less than 1, admits a finite-dimensional contractive realization [Formula presented]. Here DP is defined by the inequality [Formula presented] where P (z) is a direct sum of matrix polynomials Pi(z) (so that an appropriate Archimedean condition is satisfied), and [Formula presented], with some k-tuple n of multiplicities ni; 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 DP is a factor of det(I − KP(z)n), with a contractive matrix K.

AB - We prove that every matrix-valued rational function F, which is regular on the closure of a bounded domain DP in Cd and which has the associated Agler norm strictly less than 1, admits a finite-dimensional contractive realization [Formula presented]. Here DP is defined by the inequality [Formula presented] where P (z) is a direct sum of matrix polynomials Pi(z) (so that an appropriate Archimedean condition is satisfied), and [Formula presented], with some k-tuple n of multiplicities ni; 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 DP is a factor of det(I − KP(z)n), with a contractive matrix K.

KW - Classical Cartan domains

KW - Contractive realization

KW - Determinantal representation

KW - Multivariable polynomial

KW - Polynomially defined domain

KW - Stable polynomial

UR - http://www.scopus.com/inward/record.url?scp=85006804551&partnerID=8YFLogxK

U2 - 10.1007/978-3-319-31383-2_7

DO - 10.1007/978-3-319-31383-2_7

M3 - Article

AN - SCOPUS:85006804551

SN - 0255-0156

VL - 255

SP - 123

EP - 136

JO - Operator Theory: Advances and Applications

JF - Operator Theory: Advances and Applications

ER -