Skip to main content
SpringerLink
Log in

Strongly Regular J-inner Matrix-valued Functions and Inverse Problems for Canonical Systems

  • Conference paper
Recent Advances in Operator Theory and its Applications

Part of the book series: Operator Theory: Advances and Applications ((OT,volume 160))

Abstract

This paper provides an introduction to the role of strongly regular J-inner matrix-valued functions in the analysis of inverse problems for canonical integral and differential systems. A number of the main results that were developed in a series of papers by the authors are surveyed and examples and applications are presented, including an application to the matrix Schrödinger equation. The approach of M.G. Krein to inverse problems is discussed briefly.

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

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 129.00
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Hardcover Book
USD 169.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. D. Alpay and H. Dym, Hilbert spaces of analytic functions, inverse scattering and operator models, I, Integral Equations Operator Theory 7 (1984) 589–641.

    Article  MathSciNet  MATH  Google Scholar 

  2. D. Alpay and H. Dym, Hilbert spaces of analytic functions, inverse scattering and operator models, II, Integral Equations Operator Theory 8 (1985) 145–180.

    Article  MathSciNet  MATH  Google Scholar 

  3. D. Alpay and I. Gohberg, Inverse spectral problems for differential operators with rational scattering matrix functions, J. Differential Equations 118 (1995) 1–19.

    Article  MathSciNet  MATH  Google Scholar 

  4. D. Alpay and I. Gohberg, Inverse problems associated to a canonical differential system, in: Recent Advances in Operator Theory and Related Topics (L. Kérchy, C. Foias, I. Gohberg and H. Langer, eds.), Oper. Theor. Adv. Appl. 127, Birkhäuser, Basel, 2001, pp. 1–27.

    Google Scholar 

  5. D.Z. Arov, The generalized bitangent Carathéodory-Nevanlinna-Pick problem and (j, J 0)-inner matrix-valued functions, Russian Acad. Sci. Izvestija 42 (1994), 1–26.

    MathSciNet  Google Scholar 

  6. D.Z. Arov, On monotone families of J-contractive matrix functions, Algebra I Analiz 9 (1997), No. 6, 3–37; English transl. St. Petersburg Math. J. 9 (1998), No. 6, 1025–1051.

    MathSciNet  MATH  Google Scholar 

  7. D.Z. Arov and H. Dym, J-inner matrix functions, interpolation and inverse problems for canonical systems, I: Foundations, Integral Equations Operator Theory 29 (1997), No. 4, 373–454.

    Article  MathSciNet  MATH  Google Scholar 

  8. N. Aronszajn, Theory of reproducing kernels, Trans. Amer. Math. Soc., 68(1950), 337–404.

    MathSciNet  MATH  Google Scholar 

  9. D.Z. Arov and H. Dym, On three Krein extension problems and some generalizations, Integral Equations Operator Theory 31 (1998) 1–91.

    MathSciNet  MATH  Google Scholar 

  10. D.Z. Arov and H. Dym, J-inner matrix functions, interpolation and inverse problems for canonical systems, II: The inverse monodromy problem, Integral Equations Operator Theory 36 (2000), No. 1, 11–70.

    Article  MathSciNet  MATH  Google Scholar 

  11. D.Z. Arov and H. Dym, J-inner matrix functions, interpolation and inverse problems for canonical systems, III: More on the inverse monodromy problem, Integral Equations Operator Theory 36 (2000), No. 2, 127–181.

    Article  MathSciNet  MATH  Google Scholar 

  12. D.Z. Arov and H. Dym, Matricial Nehari problems, J-inner matrix functions and the Muckenhoupt condition, J. Funct. Anal. 181 (2001) 227–299.

    Article  MathSciNet  MATH  Google Scholar 

  13. D.Z. Arov and H. Dym, J-inner matrix functions, interpolation and inverse problems for canonical systems, IV: Direct and inverse bitangential input scattering problems, Integral Equations Operator Theory 43 (2002), No. 1, 1–67.

    MathSciNet  Google Scholar 

  14. D.Z. Arov and H. Dym, J-inner matrix functions, interpolation and inverse problems for canonical systems, V: The inverse input scattering problem for Wiener class and rational p × q input scattering matrices, Integral Equations Operator Theory 43 (2002), No. 1, 68–129.

    MathSciNet  Google Scholar 

  15. D.Z. Arov and H. Dym, Criteria for the strong regularity of J-inner functions and γ-generating matrices, J. Math. Anal. Appl. 280 (2003) 387–399.

    Article  MathSciNet  MATH  Google Scholar 

  16. D.Z. Arov and H. Dym, The bitangential inverse input impedance problem for canonical systems, I.: Weyl-Titchmarsh classification, existence and uniqueness, Integral Equations Operator Theory 47 (2003) 3–49.

    Article  MathSciNet  MATH  Google Scholar 

  17. D.Z. Arov and H. Dym, Strongly regular J-inner matrix functions and related problems, in: Current Trends in Operator Theory and its Applications (J.A. Ball, J.W. Helton, M. Klaus and L. Rodman, eds.), Oper. Theor. Adv. Appl., 149, Birkhäuser, Basel, 2004, pp. 79–106.

    Google Scholar 

  18. D.Z. Arov and H. Dym, The bitangential inverse spectral problem for canonical systems, J. Funct. Anal., 214 (2004), 312–385.

    Article  MathSciNet  MATH  Google Scholar 

  19. D.Z. Arov and H. Dym, The bitangential inverse input impedance problem for canonical systems, II.: Formulas and examples, Integral Equations and Operator Theory 51(2), 155–213 (2005).

    Article  MathSciNet  MATH  Google Scholar 

  20. D.Z. Arov and H. Dym, Direct and inverse problems for differential systems connected with Dirac systems and related factorization problems, in preparation.

    Google Scholar 

  21. L. de Branges, Some Hilbert spaces of analytic functions I, Trans. Amer. Math. Soc. 106 (1963) 445–668.

    MathSciNet  MATH  Google Scholar 

  22. L. de Branges, Hilbert Spaces of Entire Functions, Prentice-Hall, Englewood Cliffs, 1968.

    MATH  Google Scholar 

  23. L. de Branges, The expansion theorem for Hilbert spaces of entire functions, in: Entire Functions and Related Parts of Analysis, Amer. Math. Soc., Providence, 1968, pp. 79–148.

    Google Scholar 

  24. M.S. Brodskii, Triangular and Jordan Representations of Linear Operators, Transl. of Math. Monographs, 32, Amer. Math. Soc., Providence, 1971.

    Google Scholar 

  25. M.S. Brodskii and M.S. Livsic, Spectral analysis of non-selfadjoint operators and intermmediate systems, Amer. Math. Soc. Transl. (2) 13 (1960) 265–346.

    MathSciNet  Google Scholar 

  26. S. Clark and F. Gesztesy, Weyl-Titchmarsh M-function asymptotics, local uniqueness results, trace formulas, and Borg-type theorems for Dirac operators, Trans. Amer. Math. Soc. 354 (2002), 3475–3534.

    Article  MathSciNet  MATH  Google Scholar 

  27. S. Clark and F. Gesztesy, Weyl-Titchmarsh M-function asymptotics for matrix-valued Schrödinger operators, Proc. London Math. Soc. 82 (2001), 701–724.

    Article  MathSciNet  MATH  Google Scholar 

  28. H. Dym, An introduction to de Branges spaces of entire functions with applications to differential equations of Sturm-Liouville type, Advances in Math., 5 (1970), 395–471.

    MathSciNet  MATH  Google Scholar 

  29. H. Dym, J Contractive Matrix Functions, Reproducing Kernel Hilbert Spaces and Interpolation, CBMS Regional Conference Series, number 71, Amer. Math. Soc., Providence, R.I., 1989.

    Google Scholar 

  30. H. Dym, On reproducing kernels and the covariance extension problem, in: Analysis and Partial Differential Equations (C. Sadosky, ed.), Marcel Dekker, New York, 1990, pp. 427–482.

    Google Scholar 

  31. H. Dym and A. Iacob, Positive definite extensions, canonical equations and inverse problems, in: Topics in Operator Theory, Systems and Networks (H. Dym and I. Gohberg, eds.), Oper. Theory Adv. Appl. 12, Birkhäuser, Basel, 1984, pp. 141–240.

    Google Scholar 

  32. H. Dym and N. Kravitsky, On recovering the mass distribution function of a string from its spectral function, in: Topics in Functional Analysis (I. Gohberg and M. Kac, eds.), Academic Press, New York, 1978, pp. 45–90.

    Google Scholar 

  33. H. Dym and H.P. McKean, Gaussian Processes, Function Theory, and the Inverse Spectral Problem, Academic Press, New York, 1976.

    MATH  Google Scholar 

  34. F. Gesztesy, A. Kiselev and K.A. Makarov, Uniqueness results for matrix valued Schrodinger, Jacobi and Dirac-type operators, Math. Nachr. 239/240 (2002) 103–145.

    Article  MathSciNet  Google Scholar 

  35. F. Gesztesy and B. Simon, A new approach to inverse spectral theory, II. General real potentials and the connection to the spectral measure, Ann. Math., 152 (2000), 593–643.

    Article  MathSciNet  MATH  Google Scholar 

  36. I. Gohberg, M.A. Kaashoek and A.L. Sakhnovich, Canonical systems with rational spectral densities: Explicit formulas and applications, Math. Nachr. 149 (1998) 93–125.

    MathSciNet  Google Scholar 

  37. I. Gohberg, M.A. Kaashoek and A.L. Sakhnovich, Scattering problems with a pseudo-exponential potential, Asympt. Anal. 29(2002), no. 1, 1–38.

    MathSciNet  MATH  Google Scholar 

  38. I. Gohberg and M.G. Krein, Theory and applications of Volterra operators in Hilbert space, Trans. Math. Monographs, 24, Amer. Math. Soc., Providence, R.I., 1970.

    Google Scholar 

  39. M.L. Gorbachuk and V.I. Gorbachuk, M.G. Krein’s Lectures on Entire Operators, Operator Theory: Advances and Applications, 97, Birkhäuser, Basel, 1997.

    Google Scholar 

  40. A. Iacob, On the Spectral Theory of a Class of Canonical Systems of Differential Equations, PhD Thesis, The Weizmann Institute of Science, Rehovot, Israel, 1986.

    Google Scholar 

  41. I.S. Kats, ”Linear relations generated by the canonical differential equation of phase dimension 2, and eigenfunction expansions, St. Petersburg Math. J. 14 (2003), no.-3, 429–452.

    MathSciNet  Google Scholar 

  42. I.S. Kac and M.G. Krein, On the spectral functions of the string, Transl. (2) Amer. Math. Soc., 103(1974), 19–102.

    MATH  Google Scholar 

  43. M.G. Krein, On the logarithm of an infinitely decomposable Hermite-positive function, Dokl. Akad. Nauk SSSR 45 (1944), no. 3, 91–94.

    MathSciNet  MATH  Google Scholar 

  44. M.G. Krein, A contribution to the theory of entire functions of exponential type, Izv. Akad. Nauk SSSR 11 (1947) 309–326.

    MathSciNet  MATH  Google Scholar 

  45. M.G. Krein, On the theory of entire matrix functions of exponential type, Ukrain. Mat. Zh. 3 (1951), no. 2, 154–173.

    Google Scholar 

  46. M.G. Krein, Continuous analogs of theorems on polynomials orthogonal on the unit circle, Dokl. Akad. Nauk SSSR 105 (1955) 433–436.

    MathSciNet  MATH  Google Scholar 

  47. M.G. Krein, On the theory of accelerants and S-matrices of canonical differential systems, Dokl. Akad. Nauk 111 (1956), no. 6, 1167–1170.

    MathSciNet  MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  49. M.G. Krein and F.E. Melik-Adamyan, The matrix continual analogues of Schur and Carathéodory-Toeplitz problems, Izv. Akad. Nauk Armyan SSR, Ser. Mat. 21 (1986), no. 2, 107–141.

    MathSciNet  Google Scholar 

  50. M. Lesch and M.M. Malamud, The inverse spectral problem for first order systems on the half line, in: Differential operators and related topics, Vol. I (Odessa, 1997), Oper. Theory Adv. Appl. 117 (2000), Birkhauser, Basel, pp. 199–238.

    Google Scholar 

  51. B.M. Levitan and I.S. Sargsjan, Introduction to Spectral Theory, Transl. Math. Mon. 39, Amer. Math. Soc., Providence, 1975.

    Google Scholar 

  52. M.S. Livsic, Operators, Oscillations, Waves, Open Systems, Trans. Math. Monographs 34 Amer. Math. Soc., Providence, R.I., 1973.

    Google Scholar 

  53. M.M. Malamud, Uniqueness questions in inverse problems for systems of ordinary differential equations on a finite interval, Trans. Moscow Math. Soc. 60 (1999) 173–124.

    MathSciNet  MATH  Google Scholar 

  54. F.E. Melik-Adamyan, On the theory of matrix accelerants and spectral matrix functions of canonical differential systems, Dokl. Akad. Nauk Armyan SSR, 45 (1967), 145–151.

    MATH  Google Scholar 

  55. F.E. Melik-Adamyan, On canonical differential operators in Hilbert space, Izv. Akad. Nauk Armyan SSR, Ser. Mat. 12 (1977) 10–31.

    MATH  Google Scholar 

  56. F.E. Melik-Adamyan, Description of spectral functions for a class of differential operators, J. Contemp. Math. Anal. 34 (1999), no. 2, 54–70 (2000).

    MathSciNet  MATH  Google Scholar 

  57. F.E. Melik-Adamyan, Description of spectral functions for a class of differential operators with decaying boundary conditions, J. Contemp. Math. Anal. 34 (1999), no. 3, 64–74 (2000).

    MathSciNet  MATH  Google Scholar 

  58. F.E. Melik-Adamyan, Spectral functions of canonical differential equations, J. Contemp. Math. Anal. 35 (2000), no. 2, 42–60 (2001).

    MathSciNet  MATH  Google Scholar 

  59. S.A. Orlov, Nested matrix discs that depend analytically on a parameter and theorems on the invariance of the ranks of the radii of the limit matrix discs, Izv. Akad. Nauk. SSSR Ser. Mat. 40 (1976), No. 3, 593–644, 710.

    MathSciNet  MATH  Google Scholar 

  60. V.P. Potapov, The multiplicative structure of J-contractive matrix functions, Trudy Mosk. Mat. Obshch. 4 (1955) 125–236, English: Amer. Math. Soc. Transl. (2) 15 (1960) 131–243.

    MathSciNet  MATH  Google Scholar 

  61. A. Ramm and B. Simon, A new approach to inverse spectral theory, III. Short range potentials, J. d’Analyse Math., 80 (2000), 319–334.

    MathSciNet  MATH  Google Scholar 

  62. C. Remling, Schrödinger operators and de Branges spaces, J. Funct. Anal., 196 (2002), 323–394.

    Article  MathSciNet  MATH  Google Scholar 

  63. C. Remling, Inverse spectral theory for one dimensional Schrödinger operators: The A function, Math. Z., 245 (2003), 597–617.

    Article  MathSciNet  MATH  Google Scholar 

  64. M. Rosenblum and J. Rovnyak, Hardy Classes and Operator Theory, Dover Reprint, New York, 1977.

    Google Scholar 

  65. A.L. Sakhnovich, Spectral functions of a canonical system of order 2n, Math. USSR Sbornik, 71 (1992), No. 2, 355–369.

    MathSciNet  Google Scholar 

  66. L.A. Sakhnovich, Spectral problems on half-axis. Methods Funct. Anal. Topology 2 (1996), no. 3-4, 128–140.

    MathSciNet  MATH  Google Scholar 

  67. L.A. Sakhnovich, Spectral Theory of Canonical Differential Systems. Method of Operator Identities, Birkhäuser, Basel, 1999.

    MATH  Google Scholar 

  68. L.A. Sakhnovich, Works by M.G. Krein on inverse problems, Differential operators and related topics, Vol. I (Odessa, 1997), 59–69, Oper. Theory Adv. Appl., 117, Birkhäuser, Basel, 2000.

    Google Scholar 

  69. L.A. Sakhnovich, Spectral theory of a class of canonical differential systems, (Russian) Funktsional. Anal. i Prilozhen. 34 (2000), no. 2, 50–62, 96; translation in Funct. Anal. Appl. 34 (2000), no. 2, 119–128.

    MathSciNet  MATH  Google Scholar 

  70. B. Simon, A new approach to inverse spectral theory, I. Fundamental formalism, Ann. Math., 150 (1999), 1029–1057.

    Article  MATH  Google Scholar 

  71. Ju.L. Smul’yan, Operator Balls, translation in: Integral Equations Operator Theory, 13 (1990), No. 6, 864–882.

    MathSciNet  MATH  Google Scholar 

  72. H. Winkler, Small perturbations of canonical systems, Integral Equations Operator Theory, 38 (2000) 222–250.

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, South-Ukranian Pedagogical University, 65020, Odessa, Ukraine

    Damir Z. Arov

  2. Department of Mathematics, The Weizmann Institute of Science, Rehovot, 76100, Israel

    Harry Dym

Authors
  1. Damir Z. Arov
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Harry Dym
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. School of Mathematical Sciences, Tel Aviv University, Ramat Aviv, Israel

    Israel Gohberg

  2. Beer-Sheva

    D. Alpay  & P. A. Fuhrmann  & 

  3. Haifa

    J. Arazy  & L. E. Lerer  & 

  4. Tel Aviv

    A. Atzmon  & A. Ben-Artzi  & 

  5. Blacksburg

    J. A. Ball

  6. Bloomington

    H. Bercovici

  7. Chemnitz

    A. Böttcher  & G. Heinig  & 

  8. Athens, USA

    K. Clancey

  9. Buffalo

    L. A. Coburn

  10. Waterloo, Ontario

    K. R. Davidson

  11. College Station

    R. G. Douglas

  12. Groningen

    A. Dijksma

  13. Rehovot

    H. Dym

  14. Mainz

    B. Gramsch

  15. La Jolla

    J. A. Helton

  16. Amsterdam

    M. A. Kaashoek

  17. Argonne

    H. G. Kaper

  18. Tokyo

    S. T. Kuroda

  19. Calgary

    P. Lancaster

  20. Columbus

    B. Mityagin

  21. Manhattan, Kansas

    V. V. Peller

  22. Williamsburg

    L. Rodman  & I. M. Spitkovsky  & 

  23. Charlottesville

    J. Rovnyak

  24. Berkeley

    D. E. Sarason  & D. Voiculescu  & 

  25. Providence

    S. Treil

  26. Marburg

    H. Upmeier

  27. Leiden

    S. M. Verduyn Lunel

  28. Santa Cruz

    H. Widom

  29. Nashville

    D. Xia

  30. Rennes

    D. Yafaev

  31. Department of Mathematics, FEW Vrije Universiteit, De Boelelaan 1081A, 1081 HV, Amsterdam, The Netherlands

    Marinus A. Kaashoek

  32. Dipartimento di Matematica, Università di Cagliari, Viale Merello 92, 09123, Cagliari, Italy

    Sebastiano Seatzu  & Cornelis van der Mee  & 

Additional information

To Israel Gohberg, valued teacher, colleague and friend, on his 75th birthday.

Rights and permissions

Reprints and permissions

Copyright information

© 2005 Birkhäuser Verlag Basel/Switzerland

About this paper

Cite this paper

Arov, D.Z., Dym, H. (2005). Strongly Regular J-inner Matrix-valued Functions and Inverse Problems for Canonical Systems. In: Gohberg, I., et al. Recent Advances in Operator Theory and its Applications. Operator Theory: Advances and Applications, vol 160. Birkhäuser Basel. https://doi.org/10.1007/3-7643-7398-9_6

Download citation

Publish with us

Policies and ethics

Search

Navigation