Skip to main content
SpringerLink
Log in
Book cover

Operator Theory pp 1–46Cite as

Schur Analysis in an Indefinite Setting

  • Living reference work entry
  • First Online:

  • 328 Accesses

Abstract

Schur analysis comprises topics like: the Schur transformation on the class of Schur functions (by definition, the functions which are holomorphic and bounded by 1 on the open unit disk) and the Schur algorithm, Schur parameters and approximation, interpolation problems for Schur functions, factorization of rational 2 × 2 matrix polynomials, which are \(\left [\begin{array}{*{10}c} 1& 0\\ 0 &-1 \end{array} \right ]\)-unitary on the unit circle, and a related inverse scattering problem. This note contains a survey of indefinite versions of these topics related to the class of scalar generalized Schur functions. These are the meromorphic functions s(z) on the open unit disk for which the kernel \(\frac{1-s(z)s(w)^{{\ast}}} {1-zw^{{\ast}}}\) has finitely many negative squares. We also review a generalization of the Schur transformation to classes of functions on a general domain one of which is the class of scalar generalized Nevanlinna functions. These are the meromorphic functions n (z) on the open upper half plane for which the kernel \(\frac{n(z)-n(w)^{{\ast}}} {z-w^{{\ast}}}\) has finitely many negative squares.

This is a preview of subscription content, log in via an institution.

References

  1. Achieser, N.I.: Vorlesungen über Approximationstheorie, Mathematische Lehrbücher, Band II. Akademie, Berlin (1953)

    Google Scholar 

  2. Akhiezer, N.I.: The Classical Moment Problem. Hafner, New York (1965)

    MATH  Google Scholar 

  3. Akhiezer, N.I.: On a minimum problem in function theory and the number of roots of an algebraic equation inside the unit disk, Izv. Akad. Nauk SSSR Mat. Estestv. Nauk No. 9 (1930), 1169–1189 (Russian); English translation: Oper. Theory: Adv. Appl., vol. 95, pp. 19–35. Birkhäuser, Basel (1997)

    Google Scholar 

  4. Alpay, D.: The Schur Algorithm, Reproducing Kernel Spaces and System Theory. SMF/AMS texts and monographs, vol. 5. American Mathematical Society, Providence (1998)

    Google Scholar 

  5. Alpay, D., Bolotnikov, V.: Classes of pairs of meromorphic functions generated by nonnegative kernels and associated Nevanlinna-Pick problem. Int. Equ. Oper. Theory 23, 245–267 (1995)

    Article  MATH  MathSciNet  Google Scholar 

  6. Alpay, D., Dym, H.: Hilbert spaces of analytic functions, inverse scattering and operator models, I. Int. Equ. Oper. Theory 7, 589–640 (1984)

    Article  MATH  MathSciNet  Google Scholar 

  7. Alpay, D., Dym, H.: On Applications of Reproducing Kernel Spaces to the Schur Algorithm and Rational J -unitary Factorization. Oper. Theory: Adv. Appl., vol. 18, pp. 89–159. Birkhäuser, Basel (1986)

    Google Scholar 

  8. Alpay, D., Dym, H.: On Reproducing Kernel Spaces, the Schur Algorithm and Interpolation in a General Class of Domains. Oper. Theory: Adv. Appl., vol. 59, pp. 30–77. Birkhäuser, Basel (1992)

    Google Scholar 

  9. Alpay, D., Dym, H.: On a new class of structured reproducing kernel spaces. J. Funct. Anal. 111, 1–28 (1993)

    Article  MATH  MathSciNet  Google Scholar 

  10. Alpay, D., Dym, H.: On a new class of reproducing kernel spaces and a new generalization of the Iohvidov’s laws. Linear Algebra Appl. 178, 109–183 (1993)

    Article  MATH  MathSciNet  Google Scholar 

  11. Alpay, D., Dym, H.: On a new class of realization formulas and their appolications. Linear Algebra Appl. 241–243, 3–84 (1996)

    Article  MathSciNet  Google Scholar 

  12. Alpay, D., Tsekanovskii, E.: Subclasses of Herglotz–Nevanlinna matrix-valued functions and linear systems. Discrete Contin. Dyn. Syst. 1, 1–13. (2001)

    Article  MathSciNet  Google Scholar 

  13. Alpay, D., Dijksma, A., Rovnyak, J., de Snoo, H.: Schur Functions, Operator Colligations and Reproducing Kernel Pontryagin Spaces. Oper. Theory: Adv. Appl., vol. 96. Birkhäuser, Basel (1997)

    Google Scholar 

  14. Alpay, D., Azizov, T.Y., Dijksma, A., Langer, H.: The Schur Algorithm for Generalized Schur Functions I: Coisometric Realizations. Oper. Theory: Adv. Appl., vol. 129, pp. 1–36. Birkhäuser, Basel (2001)

    Google Scholar 

  15. Alpay, D., Azizov, T.Y., Dijksma, A., Langer, H., Wanjala, G.: Characteristic functions of coisometric colligations and fractional linear transformations arising from a basic interpolation problem, Department of Mathematics, University of Groningen, The Netherlands, Research report IWI (2002)

    Google Scholar 

  16. Alpay, D., Azizov, T.Y., Dijksma, A., Rovnyak, J.: Colligations in Pontryagin Spaces with a Symmetric Characteristic Function. Oper. Theory: Adv. Appl., vol. 130, pp. 55–82. Birkhäuser, Basel (2002)

    Google Scholar 

  17. Alpay, D., Constantinescu, T., Dijksma, A., Rovnyak, J.: Notes on Interpolation in the Generalized Schur Class I. Applications of Realization Theory. Oper. Theory: Adv. Appl., vol. 134, pp. 67–97. Birkhäuser, Basel (2002)

    Google Scholar 

  18. Alpay, D., Azizov, T.Y., Dijksma, A., Langer, H.: The Schur algorithm for generalized Schur functions II: Jordan chains and transformation of characteristic functions. Monatsh. Math. 138, 1–29 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  19. Alpay, D., Azizov, T.Y., Dijksma, A., Langer, H.: The Schur algorithm for generalized Schur functions III: J-unitary polynomials on the circle. Linear Algebra Appl. 369, 113–144 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  20. Alpay, D., Azizov, T.Y., Dijksma, A., Langer, H., Wanjala, G.: A Basic Interpolation Problem for Generalized Schur Functions and Coisometric Realizations. Oper. Theory: Adv. Appl., vol. 143, pp. 39–76. Birkhäuser, Basel (2003)

    Google Scholar 

  21. Alpay, D., Azizov, T.Y., Dijksma, A., Langer, H., Wanjala, G.: The Schur Algorithm for Generalized Schur Functions IV: Unitary Realizations. Oper. Theory: Adv. Appl., vol. 149, pp. 23–45. Birkhäuser, Basel (2004)

    Google Scholar 

  22. Alpay, D., Dijksma, A., Langer, H.: \(J_{\ell}\)-unitary factorization and the Schur algorithm for Nevanlinna functions in an indefinite setting. Linear Algebra Appl. 419, 675–709 (2006)

    Google Scholar 

  23. Alpay, D., Dijksma, A., Langer, H.: The Transformation of Issai Schur and Related Topics in an Indefinite Setting. Oper. Theory: Adv. Appl., vol. 176, pp. 1–98. Birkhäuser, Basel (2007)

    Google Scholar 

  24. Alpay, D., Dijksma, A., Langer, H., Shondin, Y.: The Schur transform for generalized Nevanlinna functions: interpolation and self-adjoint operator realizations. Compl. Anal. Oper. Theory 1, 169–210 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  25. Alpay, D., Dijksma, A., Langer, H.: Augmented Schur Parameters for Generalized Nevanlinna Functions and Approximation. Oper. Theory: Adv. Appl., vol. 188, pp. 1–30. Birkhäuser, Basel (2008)

    Google Scholar 

  26. Alpay, D., Dijksma, A., Langer, H., Volok, D.: Schur transformation for functions in a general class of domains. Indag. Math. (N.S.) 23(4), 921–969 (2012)

    Google Scholar 

  27. Azizov, T.Y., Iohvidov, I.S.: Foundations of the theory of linear operators in spaces with indefinite metric, Nauka, Moscow (1986, Russian); English translation: Linear Operators in Spaces with an Indefinite Metric. Wiley, New York (1989)

    Google Scholar 

  28. Azizov, T.Y., Dijksma, A., Wanjala, G.: Compressions of maximal dissipative and self-adjoint linear relations and of dilations. Linear Algebra Appl. 439, 771–792 (2013)

    Article  MATH  MathSciNet  Google Scholar 

  29. Ball, J.A., Gohberg, I., Rodman, L.: Interpolation of Rational Matrix Functions. Oper. Theory: Adv. Appl., vol. 45. Birkhäuser, Basel (1990)

    Google Scholar 

  30. Bart, H., Gohberg, I., Kaashoek, M.A.: Minimal Factorization of Matrix and Operator Functions. Oper. Theory: Adv. Appl., vol. 1. Birkhäuser, Basel (1979)

    Google Scholar 

  31. Bertin, M.J.: K-nombres de Pisot et de Salem. Acta Aritmetica 68(2), 113–131 (1994)

    MATH  MathSciNet  Google Scholar 

  32. Bertin, M.J., Decomps-Guilloux, A., Grandet-Hugot, M., Pathiaux-Delfosse, M., Schreiber, J.P.: Pisot and Salem Numbers. Birkhäuser, Basel (1992)

    Book  MATH  Google Scholar 

  33. Bolotnikov, V.: On the Carathéodory–Fejér interpolation problem for generalized Schur functions. Int. Equ. Oper. Theory 50(1), 9–41 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  34. Bolotnikov, V.: On an Interpolation Problem for Genralized Schur Functions. Oper. Theory: Adv. Appl., vol. 195, pp. 83–101. Birkhäuser, Basel (2010)

    Google Scholar 

  35. Bolotnikov, V.: A multipoint degenerate interpolation problem for generalized Schur functions. Oper. Matrices 4(2), 151–191 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  36. Bolotnikov, V.: An algorithm for finding low degree rational solutions to the Schur coefficient problem. Funct. Approx. Comment. Math. 42(1), 37–49 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  37. Bolotnikov, V.: Multipoint degenerate interpolation problem for generalized Schur functions: description of all solutions. Comp. Meth. Function Theory 11(1), 143–160 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  38. Bolotnikov, V., Kheifets, A.: On negative inertia of Pick matrices associated with generalized Schur functions. Int. Equ. Oper. Theory 56, 323–355 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  39. Carathéodory, C.: Über den Variabilitätsbereich der Koeffizienten von Potenzreihen, die gegebene Werte nicht annehmen. Math. Ann. 64, 95–115 (1907)

    Article  MATH  MathSciNet  Google Scholar 

  40. Carathéodory, C.: Über den Variabilitätsbereich der Fourierschen Konstanten von positiven harmonischen Funktionen. Rend. Circ. Mat. Palermo 32, 193–217 (1911)

    Article  MATH  Google Scholar 

  41. Carathéodory, C.: Über die Winkelderivierten von beschränkten analytische Funktionen, Sitzungsber. Preuss. Akad. Wiss. 32, 39–54 (1929)

    Google Scholar 

  42. Chamfy, C.: Fonctions méromorphes sur le cercle unité et leurs séries de Taylor. Ann. Inst. Fourier 8, 211–251 (1958)

    Article  MATH  MathSciNet  Google Scholar 

  43. Constantinescu, T., Gheondea, A.: The Schur algorithm and coefficient characterizations for generalized Schur functions. Proc. Amer. Math. Soc. 128(9), 2705–2713 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  44. Delsarte, P., Genin, Y., Kamp, Y.: Pseudo-Carathéodory functions and hermitian Toeplitz matrices. Philips J. Res. 41(1), 1–54 (1986)

    MATH  MathSciNet  Google Scholar 

  45. Dewilde, P., Dym, H.: Lossless inverse scattering for digital filters. IEEE Trans. Inf. Theory 30, 644–662 (1984)

    Article  MATH  MathSciNet  Google Scholar 

  46. Dijksma, A., Langer, H., de Snoo, H.S.V.: Selfadjoint Π κ –extensions of symmetric subspaces: an abstract approach to boundary problems with spectral parameter in the boundary conditions. Int. Equ. Oper. Theory 7(4), 459–515 (1984); Addendum. Int. Equ. Oper. Theory 7(6), 905 (1984)

    Google Scholar 

  47. Dijksma, A., Langer, H., de Snoo, H.S.V.: Hamiltonian systems with eigenvalue depending boundary conditions. Oper. Theory: Adv. Appl. 35, 37–83 (1988)

    Google Scholar 

  48. Dijksma, A., Wanjala, G.: Generalized Schur functions and augmented Schur parameters. Oper. Theory: Adv. Appl., vol. 162, pp. 135–144. Birkhäuser, Basel (2005)

    Google Scholar 

  49. Dritschel, M.A., Rovnyak, J.: Operators on indefinite inner product spaces. In: Lectures on Operator Theory and Its Applications. Fields Institute Monographs, vol. 3, pp. 143–232. American Mathematical Society, Providence (1995)

    Google Scholar 

  50. Dufresnoy, J.: Le problème des coefficients pour certaines fonctions méromorphes dans le cercle unité. Ann. Acad. Sc. Fenn. Ser. A.I 250(9), 1–7 (1958)

    MathSciNet  Google Scholar 

  51. Dufresnoy, J., Pisot, Ch.: Étude de certaines fonctions méromorphes bornes sur le cercle unité. Application à un ensemble fermé d’entiers algébriques. Annales scientifiques de l’É.N.S. 3e série 72(1), 69–92 (1955)

    Google Scholar 

  52. Dym, H.: J Contractive Matrix Functions, Reproducing Kernel Hilbert Spaces and Interpolation, CBMS Regional Conference Series, vol. 71. American Mathematical Society, Providence (1989)

    Google Scholar 

  53. Dym, H., Katsnelson, V.: Contributions of Issai Schur to Analysis. Studies in memory of Issai Schur (Chevaleret/Rehovot, 2000), xci–clxxxviii. Progr. Math., vol. 210. Birkhäuser, Boston (2003)

    Google Scholar 

  54. Fritzsche, B., Kirstein, B. (eds.): Ausgewählte Arbeiten zu den Ursprüngen der Schur-Analysis, Band 16, B.G. Teubner Verlagsgesellschaft, Leipzig (1991)

    Google Scholar 

  55. Fritzsche, B., Katsnelson, V., Kirstein, B.: The Schur Algorithm in Terms of System Realizations. Oper. Theory: Adv. Appl., vol. 197, pp. 181–250. Birkhäuser, Basel (2009)

    Google Scholar 

  56. Golinskii, L.B.: A generalization of the Nevanlinna-Pick matrix problem, Izv. Akad. Nauk Armyan SSR Ser. Mat. 18, 187–205 (1983) (Russian). English translation: Soviet J. Contemporary Math. Anal. 18(3), 22–39 (1983)

    Google Scholar 

  57. Herglotz, G.: Über Potenzreihen mit positivem, reellen Teil im Einheitskreis, Sitzungsber. Sächs. Akad. Wiss. Leipzig, Math. Phys. Kl. 63, 501–511 (1911)

    Google Scholar 

  58. Iokhvidov, I.S.: Hankel and Toeplitz Matrices and Forms. Algebraic Theory. Birkhäuser, Basel (1982)

    MATH  Google Scholar 

  59. Iohvidov, I.S., Krein, M.G., Langer, H.: Introduction to the Spectral Theory of Operators in Spaces with an Indefinite Metric. Akademie, Berlin (1982)

    MATH  Google Scholar 

  60. Kaltenbäck, M., Woracek, H.: Unique prime factorization in a partial semigroup of matrix-polynomials. Discuss. Math. Gen. Algebra Appl. 26(1), 21–43 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  61. Krein, M.G., Langer, H.: Über die verallgemeinerten Resolventen und die charakteristische Funktion eines isometrischen Operators im Raume Π k, Colloquia Math. Soc. Janos Bolyai, Tihany (Hungary), Hilbert space operators and operator algebras, vol. 5, pp. 353–399. North-Holland, Amsterdam (1972)

    Google Scholar 

  62. Krein, M.G., Langer, H.: Defect subspaces and generalized resolvents of an Hermitian operator in the space Π κ . Funct. Anal. Appl. 5, 136–146, 217–228 (1971/1972)

    Google Scholar 

  63. Krein, M.G., Langer, H.: Über einige Fortsetzungsprobleme, die eng mit der Theorie hermitescher Operatoren im Raume π k zusammenhängen. I. Einige Funktionenklassen und ihre Darstellungen. Math. Nachr. 77, 187–236 (1977)

    Article  MATH  MathSciNet  Google Scholar 

  64. Krein, M.G., Langer, H.: Über einige Fortsetzungsprobleme, die eng mit der Theorie hermitescher Operatoren im Raume Π κ zusammenhängen. II. Verallgemeinerte Resolventen, u-Resolventen und ganze Operatoren. J. Funct. Anal. 30(3), 390–447 (1978)

    Google Scholar 

  65. Krein, M.G., Langer, H.: On some extension problems which are closely connected with the theory of Hermitian operators in a space π κ . III. Indefinite analogues of the Hamburger and Stieltjes moment problems. Part I. Beiträge Anal. 14, 25–40 (1979); Part II, Beiträge Anal. 15, 27–45 (1980)

    Google Scholar 

  66. Krein, M.G., Langer, H.: On some continuation problems which are closely related to the theory of operators in spaces Π κ . IV: Continuous analogues of orthogonal polynomials on the unit circle with respect to an indefinite weightand related continuation problems for some classes of functions. J. Oper. Theory 13, 299–417 (1985)

    MATH  MathSciNet  Google Scholar 

  67. Krein, M.G., Langer, H.: Continuation of Hermitian positive definite functions and related questions. Int. Equ. Oper. Theory 78, 1–69 (2014)

    Article  MATH  MathSciNet  Google Scholar 

  68. Nevanlinna, R.: Über beschränkte Funktionen, die in gegebenen Punkten vorgeschriebene Werte annehmen. Ann. Acad. Sci. Fenn. A13(1), 1–71 (1919)

    MathSciNet  Google Scholar 

  69. Nevanlinna, R.: Asymptotische Entwicklungen beschränkter Funktionen und das Stieltjes Momentenproblem. Ann. Acad. Sci. Fenn. A18(1), 1–53 (1922)

    MathSciNet  Google Scholar 

  70. Nevanlinna, R.: Über beschränkte analytische Funktionen. Ann. Acad. Sci. Fenn. A32(7), 1–75 (1929)

    Google Scholar 

  71. Pick, G.: Über die Beschränkungen analytischer Funktionen, welche durch vorgeschriebene Funktionswerte bewirkt werden. Math. Ann. 77, 7–23 (1916)

    Article  Google Scholar 

  72. Pick, G.: Über die Beschränkungen analytischer Funktionen durch vorgegebenene Funktionswerte. Math. Ann. 78, 270–275 (1918)

    Article  MathSciNet  Google Scholar 

  73. Pick, G.: Über beschränkte Funktionen mit vorgegebene Wertzuordnungen. Ann. Acad. Sci. Fenn. A15, 3, 1–17 (1920)

    Google Scholar 

  74. Schur, I.: Über die Potenzreihen, die im Innern des Einheitskreises beschränkt sind, J. Reine Angew. Math. 147, 205–232 (1917); English translation In: I. Schur Methods in Operator Theory and Signal Processing. Oper. Theory: Adv. Appl., vol. 18, pp. 31–59. Birkhäuser, Basel (1989)

    Google Scholar 

  75. Schur, I.: Über die Potenzreihen, die im Innern des Einheitskreises beschränkt sind; Fortsetzung, J. Reine Angew. Math. 148, 122–145 (1918); English translation In: I. Schur Methods in Operator Theory and Signal Processing. Oper. Theory: Adv. Appl., vol. 18, pp. 61–88. Birkhäuser, Basel (1989)

    Google Scholar 

  76. Takagi, T.: On an algebraic problem related to an anlytic theorem of Carathéodory and Fejér and on an allied theorem of Landau. Jpn. J. Math. 1, 83–93 (1924)

    Google Scholar 

  77. Takagi, T.: Remarks on an algebraic problem. Jpn. J. Math. 2, 13–17 (1925)

    MATH  Google Scholar 

  78. Wanjala, G.: Closely Connected Unitary Realizations of the Solutions to the Basic Interpolation Problem for Generalized Schur Functions. Oper. Theory: Adv. Appl., vol. 160, pp. 441–468. Birkhäuser, Basel (2005)

    Google Scholar 

  79. Wanjala, G.: The Schur Transform of a Generalized Schur Function and Operator Realizations, Ph.D. Thesis, University of Groningen (2005)

    Google Scholar 

  80. Zeilberger, D., Dym, H.: Further properties of discrete analytic functions. J. Math. Anal. Appl. 58, 405–418 (1977)

    Article  MATH  MathSciNet  Google Scholar 

Download references

Acknowledgements

The author wants to thank the co-authors of the joint papers [14–16, 18–21] and [13, 17, 22–26] on Schur analysis for many years of pleasant cooperation, in particular Prof. Heinz Langer and Prof. Daniel Alpay.

Author information

Authors and Affiliations

  1. Johann Bernoulli Institute of Mathematics and Computer Science, University of Groningen, Groningen, The Netherlands

    Aad Dijksma

Authors
  1. Aad Dijksma
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Aad Dijksma .

Editor information

Editors and Affiliations

  1. Ben-Gurion University of the Negev, Department of Mathematics, Beer-Sheva, Israel

    Daniel Alpay

Rights and permissions

Reprints and permissions

Copyright information

© 2014 Springer Basel

About this entry

Cite this entry

Dijksma, A. (2014). Schur Analysis in an Indefinite Setting. In: Alpay, D. (eds) Operator Theory. Springer, Basel. https://doi.org/10.1007/978-3-0348-0692-3_41-1

Download citation

  • DOI: https://doi.org/10.1007/978-3-0348-0692-3_41-1

  • Received:

  • Accepted:

  • Published:

  • Publisher Name: Springer, Basel

  • Online ISBN: 978-3-0348-0692-3

  • eBook Packages: Springer Reference MathematicsReference Module Computer Science and Engineering

Publish with us

Policies and ethics

Search

Navigation