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

This article is cited in 24 scientific papers (total in 24 papers)

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
Received: 03.06.1974

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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    This publication is cited in the following articles:
    1. I. V. Kovalishina, “Analytic theory of a class of interpolation problems”, Math. USSR-Izv., 22:3 (1984), 419–463  mathnet  crossref  mathscinet  zmath
    2. V. K. Dubovoi, “Multiplicities of one-sided shifts contained in a contraction operator”, Funct. Anal. Appl., 17:1 (1983), 55–56  mathnet  crossref  mathscinet  zmath  isi
    3. D.B Hinton, J.K Shaw, “Hamiltonian systems of limit point or limit circle type with both endpoints singular”, Journal of Differential Equations, 50:3 (1983), 444  crossref
    4. D. B. Hinton, A. Schneider, “On the Titchmarsh-Weyl Coefficients for SingularS-Hermitian Systems I”, Math Nachr, 163:1 (1993), 323  crossref  mathscinet  zmath  isi
    5. Christian Remling, “Geometric Characterization of Singular Self-Adjoint Boundary Conditions for Hamiltonian Systems”, Applicable Analysis, 60:1-2 (1996), 49  crossref
    6. Fritz Gesztesy, Eduard Tsekanovskii, “On Matrix-Valued Herglotz Functions”, Math Nachr, 218:1 (2000), 61  crossref  mathscinet  zmath
    7. Gesztesy F. Kiselev A. Makarov K., “Uniqueness Results for Matrix-Valued Schrodinger, Jacobi, and Dirac-Type Operators”, Math. Nachr., 239 (2002), 103–145  crossref  mathscinet  zmath  isi
    8. E. Andersson, “On the M-function and Borg–Marchenko theorems for vector-valued Sturm–Liouville equations”, J Math Phys (N Y ), 44:12 (2003), 6077  crossref  mathscinet  zmath  adsnasa  isi
    9. Yu. M. Dyukarev, “On the indeterminacy of interpolation problems for Nevanlinna functions”, Russian Math. (Iz. VUZ), 48:8 (2004), 24–36  mathnet  mathscinet  zmath  elib
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    11. Steve Clark, Fritz Gesztesy, “On Weyl–Titchmarsh theory for singular finite difference Hamiltonian systems”, Journal of Computational and Applied Mathematics, 171:1-2 (2004), 151  crossref
    12. Yu. M. Dyukarev, “Indeterminacy of interpolation problems in the Stieltjes class”, Sb. Math., 196:3 (2005), 367–393  mathnet  crossref  crossref  mathscinet  zmath  isi  elib
    13. V. I. Khrabustovsky, “On the characteristic operators and projections and on the solutions of Weyl type of dissipative and accumulative operator systems. I. General case”, Zhurn. matem. fiz., anal., geom., 2:2 (2006), 149–175  mathnet  mathscinet  zmath  elib
    14. Yu. M. Dyukarev, I. Yu. Serikova, “Complete indeterminacy of the Nevanlinna–Pick problem in the class $S[a,b]$”, Russian Math. (Iz. VUZ), 51:11 (2007), 17–29  mathnet  crossref  mathscinet  elib
    15. D. B. Hinton, A. Schneider, “On the Titchmarsh-Weyl Coefficients for Singular S-Hermitian Systems II”, Math Nachr, 185:1 (2009), 67  crossref
    16. V. I. Khrabustovskyi, “On the limit of regular dissipative and self-adjoint boundary value problems with nonseparated boundary conditions when an interval stretches to the semiaxis”, Zhurn. matem. fiz., anal., geom., 5:1 (2009), 54–81  mathnet  mathscinet  elib
    17. Hermann Schulz-Baldes, “Geometry of Weyl theory for Jacobi matrices with matrix entries”, J Anal Math, 110:1 (2010), 129  crossref
    18. Stephen Clark, Petr Zemánek, “On a Weyl–Titchmarsh theory for discrete symplectic systems on a half line”, Applied Mathematics and Computation, 217:7 (2010), 2952  crossref
    19. V. M. Bruk, “On linear relations generated by a differential expression and by a Nevanlinna operator function”, Zhurn. matem. fiz., anal., geom., 7:2 (2011), 115–140  mathnet  mathscinet  zmath  elib
    20. V. M. Bruk, “Linear relations generated by an integral equation with Nevanlinna operator measure”, Russian Math. (Iz. VUZ), 56:10 (2012), 1–14  mathnet  crossref  mathscinet
    21. B. Fritzsche, B. Kirstein, I. Ya. Roitberg, A. L. Sakhnovich, “Skew-Self-Adjoint Dirac System with a Rectangular Matrix Potential: Weyl Theory, Direct and Inverse Problems”, Integr. Equ. Oper. Theory, 2012  crossref
    22. V. M. Bruk, “Invertible linear relations generated by an integral equation with a Nevanlinna measure”, Russian Math. (Iz. VUZ), 57:2 (2013), 13–24  mathnet  crossref
    23. A. M. Kholkin, F. S. Rofe-Beketov, “On Spectrum of Differential Operator with Block-Triangular Matrix Coefficients”, Zhurn. matem. fiz., anal., geom., 10:1 (2014), 44–63  mathnet  crossref  mathscinet
    24. Yu. M. Dyukarev, A. E. Choque Rivero, “Criterion for the Complete Indeterminacy of the Nevanlinna–Pick Matrix Problem”, Math. Notes, 96:5 (2014), 651–665  mathnet  crossref  crossref  mathscinet  zmath  isi  elib
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