×

Contractive determinantal representations of stable polynomials on a matrix polyball. (English) Zbl 1343.15008

Let \(p\) be a polynomial in commuting indeterminates \(z_{ij}^{(r)}\), \(r=1,\dots,k\), \(i=1,\dots l_r\), \(j=1,\dots,m_r\), with \(p(0)=1\). Consider the matrix unit polyball \[ {\mathcal B}=\{(Z^{(1)},\dots,Z^{(k)})\in\mathbb{C}^{l_1\times m_1}\times\dots \times\mathbb{C}^{l_k\times m_k}:\|Z^{(r)}\|<1,r=1,\dots,k\}, \] where \(\|\cdot\|\) is the 2-operator norm. The authors prove that if \(p\) has no zeros in the closure \(\bar{\mathcal B}\), then there exist \(n=(n_1,\dots,n_k)\in\mathbb{Z}_+^k\) and \(K\in\mathbb{C}^{\sum_{r=1}^km_rn_r\times\sum_{r=1}^kl_rn_r}\), \(\|K\|<1\), so that \[ p=\det{(I-KZ_n)}, \] where \[ Z_n\in\bigoplus_{r=1}^k\,(Z^{(r)}\otimes I_{n_r}),\quad Z^{(r)}=[z_{ij}^{(r)}]\in\mathbb{C}^{l_r\times m_r}. \] In the abstract, they describe the methods of this interesting paper as follows: “This result is obtained via a noncommutative lifting and a theorem on the singularities of minimal noncommutative structured system realizations.”

MSC:

15A15 Determinants, permanents, traces, other special matrix functions
32A10 Holomorphic functions of several complex variables
47N70 Applications of operator theory in systems, signals, circuits, and control theory
14A22 Noncommutative algebraic geometry

References:

[1] Agler, J.: On the representation of certain holomorphic functions defined on a polydisc. In: Topics in Operator Theory: Ernst D. Hellinger Memorial Volume, Oper. Theory Adv. Appl., Vol. 48, pp. 47-66. Birkhäuser, Basel (1990) · Zbl 0733.32002
[2] Ambrozie, C.-G., Timotin, D.A.: Von Neumann type inequality for certain domains in \[{\mathbb{C}}^nCn\]. Proc. Am. Math. Soc. 131(3), 859-869 (2003). (electronic) · Zbl 1047.47003 · doi:10.1090/S0002-9939-02-06321-9
[3] Arov, D.Z.: Passive linear steady-state dynamical systems. Sib. Mat. Zh. 20(2), 211-228 (1979). 457 · Zbl 0414.93014 · doi:10.1007/BF00970018
[4] Ball, J.A., Bolotnikov, V.: Realization and interpolation for Schur-Agler-class functions on domains with matrix polynomial defining function in \[\mathbb{C}^nCn\]. J. Funct. Anal. 213(1), 45-87 (2004) · Zbl 1061.47014 · doi:10.1016/j.jfa.2004.04.008
[5] Ball, J.A., Groenewald, G., Malakorn, T.: Structured noncommutative multidimensional linear systems. SIAM J. Control Optim. 44(4), 1474-1528 (2005) · Zbl 1139.93006 · doi:10.1137/S0363012904443750
[6] Ball, J.A., Kaliuzhnyi-Verbovetskyi, D.S.: Rational Cayley inner Herglotz-Agler functions: positive-kernel decompositions and transfer-function realizations. Linear Algebra Appl. 456, 138-156 (2014) · Zbl 1295.32003 · doi:10.1016/j.laa.2013.10.022
[7] Basu, S., Fettweis, A.: New results on stable multidimensional polynomials. II. Discrete case. IEEE Trans. Circuits Syst. 34, 1264-1274 (1987) · Zbl 0646.94018 · doi:10.1109/TCS.1987.1086065
[8] Borcea, J., Brändén, P., Liggett, T.M.: Negative dependence and the geometry of polynomials. J. Am. Math. Soc. 22(2), 521-567 (2009) · Zbl 1206.62096 · doi:10.1090/S0894-0347-08-00618-8
[9] Doyle, J.C.: Analysis of feedback systems with structured uncertainties. Proc. IEE-D 129(6), 242-250 (1982) · doi:10.1049/ip-d.1982.0053
[10] Grinshpan, A., Kaliuzhnyi-Verbovetskyi, D.S., Vinnikov, V., Woerdeman, H.J.: Stable and real-zero polynomials in two variables. Multidimens. Syst. Sign. Process. (2014). doi:10.1007/s11045-014-0286-3 · Zbl 1375.32008 · doi:10.1007/s11045-014-0286-3
[11] Grinshpan, A., Kaliuzhnyi-Verbovetskyi, D.S., Vinnikov, V., Woerdeman, H.J.: Matrix-valued Hermitian Positivstellensatz, lurking contractions, and contractive determinantal representations of stable polynomials. Oper. Theory: Adv. Appl., to appear. Available online in arXiv:1501.05527 · Zbl 1377.47005
[12] Grinshpan, A., Kaliuzhnyi-Verbovetskyi, D.S., Woerdeman, H.J.: Norm-constrained determinantal representations of multivariable polynomials. Complex Anal. Oper. Theory 7, 635-654 (2013) · Zbl 1273.15004 · doi:10.1007/s11785-012-0262-6
[13] Gurvits, L.: Van der Waerden/Schrijver-Valiant like conjectures and stable (aka hyperbolic) homogeneous polynomials: one theorem for all. With a corrigendum. Electron. J. Comb. 15(1), Research Paper 66, 26 pp (2008) · Zbl 1182.15008
[14] Hua, L.K.: Harmonic analysis of functions of several complex variables in the classical domains. Translated from the Russian by Leo Ebner and Adam Korányi, American Mathematical Society, Providence, R.I. (1963), iv+164 · Zbl 1219.47028
[15] Kaliuzhnyi-Verbovetskyi, D.S., Vinnikov, V.: Foundations of Free Non-commutative Function Theory. Math Surveys and Monographs, Vol. 199, 183 pp. AMS (2014) · Zbl 1312.46003
[16] Kaliuzhnyi-Verbovetskyi, D.S., Vinnikov, V.: Noncommutative rational functions, their difference-differential calculus and realizations. Multidimens. Syst. Signal Process. 23(1-2), 49-77 (2012) · Zbl 1255.93073 · doi:10.1007/s11045-010-0122-3
[17] Kaliuzhnyi-Verbovetskyi, D.S., Vinnikov, V.: Singularities of rational functions and minimal factorizations: the noncommutative and the commutative setting. Linear Algebra Appl. 430(4), 869-889 (2009) · Zbl 1217.47032 · doi:10.1016/j.laa.2008.08.027
[18] Knese, G.: Rational inner functions in the Schur-Agler class of the polydisk. Publ. Mat. 55, 343-357 (2011) · Zbl 1219.47028 · doi:10.5565/PUBLMAT_55211_04
[19] Kummert, A.: 2-D stable polynomials with parameter-dependent coefficients: generalizations and new results. IEEE Trans. Circuits Syst. I: Fund. Theory Appl. 49, 725-731 (2002) · Zbl 1368.93240 · doi:10.1109/TCSI.2002.1010028
[20] Kummert, A.: Synthesis of two-dimmensional lossless \[m\] m-ports with prescribed scattering matrix. Circuits Syst. Signal Process. 8(1), 97-119 (1989) · Zbl 0672.94027 · doi:10.1007/BF01598747
[21] Kummert, A.: A parametric representation for \[k\] k-variable Schur polynomials. IEEE Trans. Circuits Syst. 37(10), 1288-1291 (1990) · Zbl 0742.93039 · doi:10.1109/31.103223
[22] Li, L., Xu, L., Lin, Z.: Stability and stabilisation of linear multidimensional discrete systems in the frequency domain. Int. J. Control 86(11), 1969-1989 (2013) · Zbl 1311.93067 · doi:10.1080/00207179.2013.823671
[23] Rudin, W.: Function Theory in Polydiscs. W. A. Benjamin, Inc., New York (1969) · Zbl 0177.34101
[24] Scheicher, M.: Robustly stable multivariate polynomials. Multidimens. Syst. Signal Process. 24(1), 23-50 (2013) · Zbl 1271.93118 · doi:10.1007/s11045-011-0153-4
[25] Taylor, J.L.: A joint spectrum for several commuting operators. J. Funct. Anal. 6, 172-191 (1970) · Zbl 0233.47024 · doi:10.1016/0022-1236(70)90055-8
[26] Taylor, J.L.: The analytic-functional calculus for several commuting operators. Acta Math. 125, 1-38 (1970) · Zbl 0233.47025 · doi:10.1007/BF02392329
[27] Wagner, D.G.: Multivariate stable polynomials: theory and applications. Bull. Am. Math. Soc. 48(1), 53-84 (2011) · Zbl 1207.32006 · doi:10.1090/S0273-0979-2010-01321-5
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.