Minimal Realizations and Determinantal Representations in the Indefinite Setting
Minimal Realizations and Determinantal Representations in the Indefinite Setting
复制标题
不定环境中的最小实现和行列式表示
DOI：
10.1007/s00020-022-02697-1
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发表时间：
2022
影响因子：
0.8
通讯作者：
Woerdeman, Hugo J.
中科院分区：
文献类型：
--
作者：
Jackson, Joshua D.;Woerdeman, Hugo J.
For a signature matrixJ, we show that a rational matrix functionM(z) that is strictlyJ-contractive on the unit circle, has a strict-contractive realization \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{bmatrix} A &{} B \\ C &{} D \end{bmatrix}$$\end{document} for an appropriate signature matrix; that is,. As an application, we use this result to show that a two variable polynomialof degree,, without roots onallows a determinantal representation \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} p(z_1, z_2) = p(0,0) \det (I_{n_1+1} - K Z), \ \ Z = z_1 I_{n_1} \oplus z_2 I_{n_2} , \end{aligned}$$\end{document}whereKis a strict-contraction. This provides first evidence of a new conjecture that a two variable polynomialof degreehas a determinantal representation withKa strict-contraction if and only ifhas no roots in.
For a signature matrixJ, we show that a rational matrix functionM(z) that is strictlyJ-contractive on the unit circle, has a strict-contractive realization \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{bmatrix} A &{} B \\ C &{} D \end{bmatrix}$$\end{document} for an appropriate signature matrix; that is,. As an application, we use this result to show that a two variable polynomialof degree,, without roots onallows a determinantal representation \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} p(z_1, z_2) = p(0,0) \det (I_{n_1+1} - K Z), \ \ Z = z_1 I_{n_1} \oplus z_2 I_{n_2} , \end{aligned}$$\end{document}whereKis a strict-contraction. This provides first evidence of a new conjecture that a two variable polynomialof degreehas a determinantal representation withKa strict-contraction if and only ifhas no roots in.
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DOI：
--
发表时间：
2013
期刊：
影响因子：
--
作者：
A. Grinshpan;Dmitry S. Kaliuzhnyi;H. Woerdeman
通讯作者：
H. Woerdeman
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DOI：
--
发表时间：
1996
期刊：
影响因子：
--
作者：
D. Alpay;V. Bolotnikov;A. Dijksma;H. Snoo
通讯作者：
H. Snoo
Kummert在bidisk上的实现方法
影响因子：
1.1
作者：
Knese, Greg
通讯作者：
Knese, Greg
Pontryagin 空间中广义 Schur 函数的最小无源实现
影响因子：
0.8
作者：
Lassi Lilleberg
通讯作者：
Lassi Lilleberg
Schur 函数、算子关联和再现核 Pontryagin 空间
影响因子：
2.1
作者：
D. Alpay;A. Dijksma;J. Rovnyak;H. de Snoo
通讯作者：
H. de Snoo