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 Izv. Akad. Nauk SSSR Ser. Mat., 1976, Volume 40, Issue 3, Pages 593–644 (Mi izv2143)

Nested matrix disks analytically depending parameter, and theorems on the invariance radii of limiting disks

S. A. Orlov

Abstract: In this work there is an investigation of a family of invertible (i.e. $W(b,\lambda)\not\equiv0$) analytic matrix-valued functions $W(b,\lambda)$ ($0<b<\infty$) which are $J$-contractive ($\Gamma(b,\lambda)\overset{\mathrm{def}}= J-W(b,\lambda)JW^*(b,\lambda)>0$, $J^*=J$, $J^2=I$) and which have monotonically increasing $J$-forms $\Gamma(b,\lambda)$ as $b\to\infty$. Invariance with respect to $\lambda$ of the rank of the matrix $R^2(\lambda)=\lim_{b\to\infty}\Gamma^{-1}(b,\lambda)$ is established, and conditions for convergence of $W(b,\lambda)$ are investigated. As a special case a theorem is obtained on the invariance of ranks of limiting radii of Weyl disks, which is fundamentally of significance in the theory of classical problems (the moment problem, the Nevanlinna–Pick problem, the Weyl problem on the number of square-integrable solutions of a system of differential equations, and so forth).
Bibliography: 17 titles.

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English version:
Mathematics of the USSR-Izvestiya, 1976, 10:3, 565–613

Bibliographic databases:

UDC: 517.5+517.9
MSC: Primary 34B20, 15A03, 15A57; Secondary 15A21, 30A80, 15A45

Citation: S. A. Orlov, “Nested matrix disks analytically depending parameter, and theorems on the invariance radii of limiting disks”, Izv. Akad. Nauk SSSR Ser. Mat., 40:3 (1976), 593–644; Math. USSR-Izv., 10:3 (1976), 565–613

Citation in format AMSBIB
\Bibitem{Orl76} \by S.~A.~Orlov \paper Nested matrix disks analytically depending parameter, and theorems on the invariance radii of limiting disks \jour Izv. Akad. Nauk SSSR Ser. Mat. \yr 1976 \vol 40 \issue 3 \pages 593--644 \mathnet{http://mi.mathnet.ru/izv2143} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=425671} \zmath{https://zbmath.org/?q=an:0337.34020} \transl \jour Math. USSR-Izv. \yr 1976 \vol 10 \issue 3 \pages 565--613 \crossref{https://doi.org/10.1070/IM1976v010n03ABEH001718} 

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Citing articles on Google Scholar: Russian citations, English citations
Related articles on Google Scholar: Russian articles, English articles

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