Skip to main content
SpringerLink
Log in

de Branges Spaces and Kreĭn’s Theory of Entire Operators

  • Reference work entry
  • First Online:
Operator Theory

Abstract

This work presents a contemporary treatment of Kreĭn’s entire operators with deficiency indices (1, 1) and de Branges’ Hilbert spaces of entire functions. Each of these theories played a central role in the research of both renown mathematicians. Remarkably, entire operators and de Branges spaces are intimately connected and the interplay between them has had an impact in both spectral theory and the theory of functions. This work exhibits the interrelation between Kreĭn’s and de Branges’ theories by means of a functional model and discusses recent developments, giving illustrations of the main objects and applications to the spectral theory of difference and differential operators.

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 999.00
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Hardcover Book
USD 549.99
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

References

  1. Akhiezer, N.I.: The classical moment problem and some related questions in analysis. Translated by N. Kemmer. Hafner Publishing, New York (1965)

    MATH  Google Scholar 

  2. Akhiezer, N.I., Glazman, I.M.: Theory of Linear Operators in Hilbert Space. Dover, New York (1993). Translated from the Russian and with a preface by Merlynd Nestell, Reprint of the 1961 and 1963 translations, Two volumes bound as one

    Google Scholar 

  3. Arens, R.: Operational calculus of linear relations. Pac. J. Math. 11, 9–23 (1961)

    Article  MathSciNet  MATH  Google Scholar 

  4. Baranov, A.: Polynomials in the de Branges spaces of entire functions. Ark. Mat. 44(1), 16–38 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  5. Berezans′kiĭ, Ju.M.: Expansions in Eigenfunctions of Selfadjoint Operators. Translated from the Russian by R. Bolstein, J.M. Danskin, J. Rovnyak, L. Shulman. Translations of Mathematical Monographs, vol. 17. American Mathematical Society, Providence (1968)

    Google Scholar 

  6. Bulla, W., Gesztesy, F.: Deficiency indices and singular boundary conditions in quantum mechanics. J. Math. Phys. 26(10), 2520–2528 (1985)

    Article  MathSciNet  MATH  Google Scholar 

  7. Birman, M.Sh., Solomjak, M.Z.: Spectral Theory of Selfadjoint Operators in Hilbert Space. Mathematics and Its Applications (Soviet Series). D. Reidel Publishing, Dordrecht (1987). Translated from the 1980 Russian original by S. Khrushchëv, V. Peller

    Google Scholar 

  8. de Branges, L.: Some Hilbert spaces of entire functions. Proc. Am. Math. Soc. 10, 840–846 (1959)

    Article  MATH  Google Scholar 

  9. de Branges, L.: Some Hilbert spaces of entire functions. Trans. Am. Math. Soc. 96, 259–295 (1960)

    Article  MATH  Google Scholar 

  10. de Branges, L.: Some Hilbert spaces of entire functions. Bull. Am. Math. Soc. 67, 129–134 (1961)

    Article  MATH  Google Scholar 

  11. de Branges, L.: Some Hilbert spaces of entire functions. II. Trans. Am. Math. Soc. 99, 118–152 (1961)

    Article  MATH  Google Scholar 

  12. de Branges, L.: Some Hilbert spaces of entire functions. III. Trans. Am. Math. Soc. 100, 73–115 (1961)

    Article  MATH  Google Scholar 

  13. de Branges, L.: Some Hilbert spaces of entire functions. IV. Trans. Am. Math. Soc. 105, 43–83 (1962)

    Article  MATH  Google Scholar 

  14. de Branges, L.: Hilbert Spaces of Entire Functions. Prentice-Hall, Englewood Cliffs (1968)

    MATH  Google Scholar 

  15. de Branges, L.: A proof of the Bieberbach conjecture. Acta Math. 154(1–2), 137–152 (1985)

    Article  MathSciNet  MATH  Google Scholar 

  16. de Branges, L., Rovnyak, J.: Canonical models in quantum scattering theory. In: Perturbation Theory and Its Applications in Quantum Mechanics (Proc. Adv. Sem. Math. Res. Center, U.S.Army, Theoret. Chem. Inst., Univ. of Wisconsin, Madison, Wis., 1965), pp. 295–392. Wiley, New York (1966)

    Google Scholar 

  17. Dijksma, A., de Snoo, H.S.V.: Self-adjoint extensions of symmetric subspaces. Pac. J. Math. 54, 71–100 (1974)

    Article  MATH  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  19. Dym, H., McKean, H.P.: Gaussian Processes, Function Theory, and the Inverse Spectral Problem. Probability and Mathematical Statistics, vol. 31. Academic [Harcourt Brace Jovanovich Publishers], New York (1976)

    Google Scholar 

  20. Eckhardt, J.: Inverse uniqueness results for Schrödinger operators using de Branges theory. Complex Anal. Oper. Theory 8, 37–50 (2014)

    Article  MathSciNet  MATH  Google Scholar 

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

    Google Scholar 

  22. Griffith, J.L.: Hankel transforms of functions zero outside a finite interval. J. Proc. R. Soc. N S W 89, 109–115 (1955/1956)

    Google Scholar 

  23. Hassi, S., de Snoo, H.S.V.: One-dimensional graph perturbations of selfadjoint relations. Ann. Acad. Sci. Fenn. Math. 22(1), 123–164 (1997)

    MathSciNet  MATH  Google Scholar 

  24. Krein, M.G.: On a remarkable class of Hermitian operators. C. R. (Dokl.) Acad. Sci. URSS (N. S.) 44, 175–179 (1944)

    Google Scholar 

  25. Krein, M.G.: On Hermitian operators whose deficiency indices are 1. C. R. (Dokl.) Acad. Sci. URSS (N. S.) 43, 323–326 (1944)

    Google Scholar 

  26. Krein, M.G.: On Hermitian operators with deficiency indices equal to one. II. C. R. (Dokl.) Acad. Sci. URSS (N. S.) 44, 131–134 (1944)

    Google Scholar 

  27. Krein, M.G.: Analytic problems and results in the theory of linear operators in Hilbert space. In: Proceedings of the International Congress of Mathematicians (Moscow, 1966), pp. 189–216. Izdat. “Mir”, Moscow (1968)

    Google Scholar 

  28. Kostenko, A., Sakhnovich, A., Teschl, G.: Inverse eigenvalue problems for perturbed spherical Schrödinger operators. Inverse Probl. 26(10), 105013, 14 (2010)

    Google Scholar 

  29. Kaltenbäck, M., Woracek, H.: Pontryagin spaces of entire functions. I. Integral Equ. Oper. Theory 33(1), 34–97 (1999)

    Article  MATH  Google Scholar 

  30. Kaltenbäck, M., Woracek, H.: Pontryagin spaces of entire functions. II. Integral Equ. Oper. Theory 33(3), 305–380 (1999)

    Article  MATH  Google Scholar 

  31. Kaltenbäck, M., Woracek, H.: Pontryagin spaces of entire functions. III. Acta Sci. Math. (Szeged) 69(1–2), 241–310 (2003)

    Google Scholar 

  32. Kaltenbäck, M., Woracek, H.: De Branges spaces of exponential type: general theory of growth. Acta Sci. Math. (Szeged) 71(1–2), 231–284 (2005)

    Google Scholar 

  33. Kaltenbäck, M., Woracek, H.: Pontryagin spaces of entire functions. IV. Acta Sci. Math. (Szeged) 72(3–4), 709–835 (2006)

    Google Scholar 

  34. Kaltenbäck, M., Woracek, H.: Pontryagin spaces of entire functions. VI. Acta Sci. Math. (Szeged) 76(3–4), 511–560 (2010)

    Google Scholar 

  35. Kaltenbäck, M., Woracek, H.: Pontryagin spaces of entire functions. V. Acta Sci. Math. (Szeged) 77(1–2), 223–336 (2011)

    Google Scholar 

  36. Langer, H., Textorius, B.: On generalized resolvents and Q-functions of symmetric linear relations (subspaces) in Hilbert space. Pac. J. Math. 72(1), 135–165 (1977)

    Article  MathSciNet  MATH  Google Scholar 

  37. Langer, M., Woracek, H.: A characterization of intermediate Weyl coefficients. Monatsh. Math. 135(2), 137–155 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  38. Martin, R.T.W.: Representation of simple symmetric operators with deficiency indices (1, 1) in de Branges space. Complex Anal. Oper. Theory 5(2), 545–577 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  39. Naboko, S.N.: Absolutely continuous spectrum of a nondissipative operator, and a functional model. I. Zap. Naučn. Sem. Leningrad. Otdel Mat. Inst. Steklov. 65, 90–102, 204–205 (1976) Investigations on linear operators and the theory of functions, VII.

    Google Scholar 

  40. Naboko, S.N.: Absolutely continuous spectrum of a nondissipative operator, and a functional model. II. Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. 73, 118–135, 232–233 (1977/1978). Investigations on linear operators and the theory of functions, VIII.

    Google Scholar 

  41. Pavlov, B.S.: Conditions for separation of the spectral components of a dissipative operator. Izv. Akad. Nauk SSSR Ser. Mat. 39, 123–148, 240 (1975)

    Google Scholar 

  42. Pavlov, B.S.: Selfadjoint dilation of a dissipative Schrödinger operator, and expansion in eigenfunctions. Funkcional. Anal. i Priložen. 9(2), 87–88 (1975)

    Google Scholar 

  43. Remling, C.: Schrödinger operators and de Branges spaces. J. Funct. Anal. 196(2), 323–394 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  44. Shmulian, Yu.L.: Representation of Hermitian operators with an ideal reference subspace. Mat. Sb. (N.S.) 85(127), 553–562 (1971)

    Google Scholar 

  45. Simon, B.: The classical moment problem as a self-adjoint finite difference operator. Adv. Math. 137(1), 82–203 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  46. Nagy, B.Sz., Foiaş, C.: Harmonic Analysis of Operators on Hilbert Space. Translated from the French and revised. North-Holland, Amsterdam (1970)

    Google Scholar 

  47. Silva, L.O., Toloza, J.H.: On the spectral characterization of entire operators with deficiency indices (1, 1). J. Math. Anal. Appl. 367(2), 360–373 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  48. Silva, L.O., Toloza, J.H.: The class of n-entire operators. J. Phys. A 46(2), 025202, 23 (2013)

    Google Scholar 

  49. Silva, L.O., Toloza, J.H.: A class of n-entire Schrödinger operators. Complex Anal. Oper. Theory 8, 1581–1599 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  50. Silva, L.O., Toloza, J.H.: On dB spaces with nondensely defined multiplication operator and the existence of zero-free functions. J. Math. Anal. Appl. 421, 996–1005 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  51. Silva, L.O., Toloza, J.H.: The spectra of selfadjoint extensions of entire operators with deficiency indices (1, 1). In: Janas, J., Kurasov, P., Laptev, A., Naboko, S. (eds.) Operator Methods in Mathematical Physics. Conference on Operator Theory, Analysis and Mathematical Physics (OTAMP) 2010, Bedlewo, Poland. Operator Theory: Advances and Applications, vol. 227, pp. 151–164. Birkhäuser/Springer, Basel (2013)

    Google Scholar 

  52. Strauss, A.V.: Functional models and generalized spectral functions of symmetric operators. Algebra i Analiz 10(5), 1–76 (1998)

    MATH  Google Scholar 

  53. Strauss, A.V.: Functional models of regular symmetric operators. In: Operator Theory and Its Applications (Winnipeg, MB, 1998). Fields Institute Communications, vol. 25, pp. 1–13. American Mathematical Society, Providence (2000)

    Google Scholar 

  54. Strauss, A.V.: Functional models of linear operators. In: Operator Theory, System Theory and Related Topics (Beer-Sheva/Rehovot, 1997). Operator Theory, Advances and Applications, vol. 123, pp. 469–484. Birkhäuser, Basel (2001)

    Google Scholar 

  55. Tsekanovskii, E.R., Shmulian, Yu.L.: Questions in the theory of the extension of unbounded operators in rigged Hilbert spaces. In: Mathematical analysis, vol. 14 (Russian), pp. 59–100, i. (loose errata). Akad. Nauk SSSR Vsesojuz. Inst. Naučn. i Tehn. Informacii, Moscow (1977)

    Google Scholar 

  56. Woracek, H.: Existence of zerofree functions N-associated to a de Branges Pontryagin space. Monatsh. Math. 162(4), 453–506 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  57. Zemanian, A.H.: The Hankel transformation of certain distributions of rapid growth. SIAM J. Appl. Math. 14, 678–690 (1966)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

Julio H. Toloza has been partially supported by CONICET (Argentina) through grant PIP 112-201101-00245.

Author information

Authors and Affiliations

  1. Departamento de Física Matemática, Instituto de Investigaciones en Matemáticas Aplicadas y en Sistemas, Universidad Nacional Autónoma de México, Mexico DF, Mexico

    Luis O. Silva

  2. Consejo Nacional de Investigaciones Científicas y Técnicas (CONICET), and Centro de Investigación en Informática para la Ingeniería, Universidad Tecnológica Nacional – Facultad Regional Córdoba, Maestro M. López s/n, Córdoba, Argentina

    Julio H. Toloza

Authors
  1. Luis O. Silva
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Julio H. Toloza
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding authors

Correspondence to Luis O. Silva or Julio H. Toloza .

Editor information

Editors and Affiliations

  1. Earl Katz Chair in Algebraic System Theory, Department of Mathematics, Ben-Gurion University of the Negev, Be’er Sheva, Israel

    Daniel Alpay

Rights and permissions

Reprints and permissions

Copyright information

© 2015 Springer Basel

About this entry

Cite this entry

Silva, L.O., Toloza, J.H. (2015). de Branges Spaces and Kreĭn’s Theory of Entire Operators. In: Alpay, D. (eds) Operator Theory. Springer, Basel. https://doi.org/10.1007/978-3-0348-0667-1_4

Download citation

Publish with us

Policies and ethics

Search

Navigation