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The Matricial Carathéodory Problem in Both Nondegenerate and Degenerate Cases

Chapter
Part of the Operator Theory: Advances and Applications book series (OT, volume 165)

Abstract

The main goal of this paper is to present a new approach to both the nondegenerate and degenerate case of the matricial Carathéodory problem. This approach is based on the analysis of central matrix-valued Carathéodory functions which was started in [FK1] and then continued in [FK3]. In the nondegenerate situation we will see that the parametrization of the solution set obtained here coincides with the well-known formula of D.Z. Arov and M.G. Krein for that case (see [AK]).

Keywords

Matricial Carathéodory problem Arov-Krein representation of the solution set central matrix-valued Carathéodory functions 

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References

  1. [Al]
    Albert, A.: Conditions for positive and nonnegative definiteness in terms of pseudoinverses, SIAM J. Appl. Math. 17 (1969), 434–440.zbMATHMathSciNetCrossRefGoogle Scholar
  2. [AK]
    Arov, D.Z.; Krein, M.G.: Problems of the search of the minimum of entropy in indeterminate extension problems (Russian), Funkcional. Anal. Prilozen. 15 (1981), No. 2, 61–64; English translation: Func. Anal. Appl. 15 (1981), 123–126.MathSciNetGoogle Scholar
  3. [BGR]
    Ball, J.A.; Gohberg, I.; Rodman, L.: Interpolation of Rational Matrix Functions, Operator Theory: Advances and Applications 45, Birkhäuser, Basel 1990.Google Scholar
  4. [BH]
    Ball, J.A.; Helton, J.W.: Interpolation problems of Pick-Nevanlinna and Loewner types for meromorphic matrix functions: parametrization of the set of all solutions, Integral Equations Operator Theory 9 (1986), 155–203.MathSciNetCrossRefzbMATHGoogle Scholar
  5. [BD]
    Bolotnikov, V.; Dym, H.: On degenerate interpolation, entropy and extremal problems for matrix Schur functions, Integral Equations Operator Theory 32 (1998), 367–435.MathSciNetCrossRefzbMATHGoogle Scholar
  6. [Br]
    Bruinsma, P.: Degenerate interpolation problems for Nevanlinna pairs, Indag. Math., N.S. 2 (1991), 179–200.zbMATHMathSciNetCrossRefGoogle Scholar
  7. [CH1]
    Chen, G.N.; Hu, Y.J.: The truncated Hamburger matrix moment problems in the nondegenerate and degenerate cases, and matrix continued fractions, Linear Algebra Appl. 277 (1998), 199–236.MathSciNetCrossRefzbMATHGoogle Scholar
  8. [CH2]
    Chen, G.N.; Hu, Y.J.: On the multiple Nevanlinna-Pick matrix interpolation in the class C p and the Carathéodory matrix coefficient problem, Linear Algebra Appl. 283 (1998), 179–203.MathSciNetCrossRefzbMATHGoogle Scholar
  9. [DGK1]
    Delsarte, P.; Genin, Y.; Kamp, Y.: Orthogonal polynomial matrices on the unit circle, IEEE Trans. Circuits and Systems CAS-25 (1978), 149–160.MathSciNetCrossRefGoogle Scholar
  10. [DGK2]
    Delsarte, P.; Genin, Y.; Kamp, Y.: Schur parameterization of positive definite block-Toeplitz systems, SIAM J. Appl. Math. 36 (1979), 34–46.MathSciNetCrossRefzbMATHGoogle Scholar
  11. [DGK3]
    Delsarte, P.; Genin, Y.; Kamp, Y.: The Nevanlinna-Pick problem for matrix-valued functions, SIAM J. Appl. Math. 36 (1979), 47–61.MathSciNetCrossRefzbMATHGoogle Scholar
  12. [Du]
    Dubovoj, V.K.: Indefinite metric in the interpolation problem of Schur for analytic matrix functions IV (Russian), Teor. Funktsii, Funkts. Anal. i Prilozen. 42 (1984), 46–57. English translation in: Topics in Interpolation Theory (Eds.: H. Dym, B. Fritzsche, V.E. Katsnelson, B. Kirstein), Operator Theory: Advances and Applications 95, Birkhäuser, Basel 1997, pp. 93–104.zbMATHGoogle Scholar
  13. [DFK]
    Dubovoj, V.K.; Fritzsche, B.; Kirstein, B.: Matricial Version of the Classical Schur Problem, Teubner-Texte zur Mathematik 129, B. G. Teubner, Stuttgart-Leipzig 1992.Google Scholar
  14. [Dy]
    Dym, H.: J Contractive Matrix Functions, Reproducing Kernel Spaces and Interpolation, CBMS Regional Conference Series in Mathematics 71, Amer. Math. Soc., Providence, R.I. 1989.Google Scholar
  15. [EP]
    Efimov, A.V.; Potapov, V.P.: J-expansive matrix-valued functions and their role in the analytic theory of electrical circuits (Russian), Uspekhi Mat. Nauk 28 (1973), 65–130; English translation: Russian Math. Surveys 28 (1973), 69–140.MathSciNetzbMATHGoogle Scholar
  16. [FF]
    Foiaş, C.; Frazho, A.E.: The Commutant Lifting Approach to Interpolation Problems, Operator Theory: Advances and Applications 45, Birkhäuser, Basel 1990.Google Scholar
  17. [FFGK]
    Foiaş, C.; Frazho, A.E.; Gohberg, I.; Kaashoek, M.A.: Metric Constrained Interpolation, Commutant Lifting and Systems, Operator Theory: Advances and Applications 100, Birkhäuser, Basel 1998.Google Scholar
  18. [FFK]
    Fritzsche, B.; Fuchs, S.; Kirstein, B.: A Schur type matrix extension problem, Part V, Math. Nachr. 158 (1992), 133–159.MathSciNetCrossRefzbMATHGoogle Scholar
  19. [FK1]
    Fritzsche, B.; Kirstein, B.: An extension problem for non-negative hermitian block Toeplitz matrices, Math. Nachr., Part I: 130 (1987), 121–135; Part II: 131 (1987), 287–297; Part III: 135 (1988), 319–341; Part IV: 143 (1989), 329–354; Part V: 144 (1989), 283–308.MathSciNetzbMATHGoogle Scholar
  20. [FK2]
    Fritzsche, B.; Kirstein, B.: A Schur type matrix extension problem, Math. Nachr. Part I: 134 (1987), 257–271; Part II: 138 (1988), 195–216; Part III: 143 (1989), 227–247; Part IV: 147 (1990), 235–258.MathSciNetzbMATHGoogle Scholar
  21. [FK3]
    Fritzsche, B.; Kirstein, B.: Representations of central matrix-valued Carathéodory functions in both nondegenerate and degenerate cases, Integral Equations Operator Theory 50 (2004), 333–361.MathSciNetCrossRefzbMATHGoogle Scholar
  22. [FKK]
    Fritzsche, B.; Kirstein, B.; Krug, V.: On several types of resolvent matrices of nondegenerate matricial Carathéodory problems, Linear Algebra Appl. 281 (1998), 137–170.MathSciNetCrossRefzbMATHGoogle Scholar
  23. [FKL]
    Fritzsche, B.; Kirstein, B.; Lasarow, A.: On rank invariance of Schwarz-Pick-Potapov block matrices of matrix-valued Carathéodory functions, in: Operator Theory: Advances and Applications 135 (Eds.: A. Böttcher, I. Gohberg, P. Junghanns), Birkhäuser, Basel 2002, pp. 161–181.Google Scholar
  24. [Ko]
    Kovalishina, I.V.: Analytic theory of a class of interpolation problems (Russian), Izv. Akad. Nauk SSSR, Ser. Mat. 47 (1983), 455–497; English translation: Math. USSR Izvestija 22 (1984), 419–463.MathSciNetGoogle Scholar
  25. [Sa]
    Sakhnovich, L.A.: Interpolation Theory and its Applications, Mathematics and its Applications 428, Kluwer, Dordrecht 1997.Google Scholar
  26. [Sc]
    Schur, I: Über Potenzreihen, die im Innern des Einheitskreises beschränkt sind, J. reine angew. Math., Part I: 147 (1917), 205–232; Part II: 148 (1918), 122–145.Google Scholar

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© Birkhäuser Verlag Basel/Switzerland 2006

Authors and Affiliations

  1. 1.Fakultät für Mathematik und InformatikUniversität LeipzigLeipzigGermany
  2. 2.Departement ComputerwetenschappenKatholieke Universiteit LeuvenHeverlee (Leuven)Belgium

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