Skip to main content
SpringerLink
Log in

On Factorizations of Analytic Operator-Valued Functions and Eigenvalue Multiplicity Questions

  • Published:
Integral Equations and Operator Theory Aims and scope Submit manuscript

An Erratum to this article was published on 22 April 2016

Abstract

We study several natural multiplicity questions that arise in the context of the Birman–Schwinger principle applied to non-self-adjoint operators. In particular, we re-prove (and extend) a recent result by Latushkin and Sukhtyaev by employing a different technique based on factorizations of analytic operator-valued functions due to Howland. Factorizations of analytic operator-valued functions are of particular interest in themselves and again we re-derive Howland’s results and subsequently extend them. Considering algebraic multiplicities of finitely meromorphic operator-valued functions, we recall the notion of the index of a finitely meromorphic operator-valued function and use that to prove an analog of the well-known Weinstein–Aronszajn formula relating algebraic multiplicities of the underlying unperturbed and perturbed operators. Finally, we consider pairs of projections for which the difference belongs to the trace class and relate their Fredholm index to the index of the naturally underlying Birman–Schwinger operator.

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

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

We’re sorry, something doesn't seem to be working properly.

Please try refreshing the page. If that doesn't work, please contact support so we can address the problem.

References

  1. Amrein W.O., Sinha K.B.: On pairs of projections in a Hilbert space. Linear Algebra Appl. 208, 425–435 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  2. Aronszajn N., Brown R.D.: Finite-dimensional perturbations of spectral problems and variational approximation methods for eigenvalue problems. Part I. Finite-dimensional perturbations. Stud. Math. 36, 1–76 (1970)

    MathSciNet  MATH  Google Scholar 

  3. Avron J., Seiler R., Simon B.: The index of a pair of projections. J. Funct. Anal. 120, 220–237 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  4. Beyn W.-J., Latushkin Y., Rottmann-Matthes J.: Finding eigenvalues of holomorphic Fredholm operator pencils using boundary value problems and contour integrals. Integr. Equ. Oper. Theory 78, 155–211 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  5. Birman M.Sh.: On the spectrum of singular boundary-value problems. Mat. Sb. (N.S.) 55(97), 125–174 (1961) (Russian). (Engl. transl. in Am. Math. Soc. Transl., Ser. 2, 53, 23–80) (1966)

  6. Birman M.Sh., Solomyak M.Z.: Estimates for the number of negative eigenvalues of the Schrödinger operator and its generalizations, in Estimates and Asymptotics for Discrete Spectra of Integral and Differential Equations (Leningrad, 1989–1990). Adv. Sov. Math. 7, 1–55 (1991)

    MathSciNet  MATH  Google Scholar 

  7. Birman M.Sh., Yafaev D.R.: The spectral shift function. The work of M. G. Krein and its further development. St. Petersb. Math. J. 4, 833–870 (1993)

    MathSciNet  Google Scholar 

  8. Böttcher A., Spitkovsky I.M.: A gentle guide to the basics of two projections theory. Lin. Algebra Appl. 432, 1412–1459 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  9. Davis C.: Separation of two linear subspaces. Acta Sci. Math. Szeged. 19, 172–187 (1958)

    MathSciNet  MATH  Google Scholar 

  10. Edmunds D.E., Evans W.D.: Spectral Theory and Differential Operators. Clarendon Press, Oxford (1989)

    MATH  Google Scholar 

  11. Effros E.: Why the circle is connected: an introduction to quantized topology. Math. Intell. 11(1), 27–35 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  12. Gesztesy F., Holden H.: A unified approach to eigenvalues and resonances of Schrödinger operators using Fredholm determinants. J. Math. Anal. Appl. 123, 181–198 (1987)

    Article  MathSciNet  MATH  Google Scholar 

  13. Gesztesy F., Latushkin Y., Makarov K.A.: Evans functions, Jost functions, and Fredholm determinants. Arch. Rat. Mech. Anal. 186, 361–421 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  14. Gesztesy F., Latushkin Y., Mitrea M., Zinchenko M.: Nonselfadjoint operators, infinite determinants, and some applications. Russ. J. Math. Phys. 12, 443–471 (2005)

    MathSciNet  MATH  Google Scholar 

  15. Gesztesy F., Latushkin Yu., Zumbrun K.: Derivatives of (modified) Fredholm determinants and stability of standing and traveling waves. J. Math. Pures Appl. 90, 160–200 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  16. Gohberg I., Goldberg S., Kaashoek M.A.: Classes of Linear Operators, Vol. I, Operator Theory: Advances and Applications, vol. 49. Birkhäuser, Basel (1990)

    Book  MATH  Google Scholar 

  17. Gohberg I., Leiterer J.: Holomorphic Operator Functions of One Variable and Applications, Operator Theory: Advances and Applications, vol. 192. Birkhäuser, Basel (2009)

    Book  MATH  Google Scholar 

  18. Gohberg I.C., Sigal E.I.: An operator generalizations of the logarithmic residue theorem and the theorem of Rouché. Math. USSR. Sbornik 13, 603–625 (1971)

    Article  MATH  Google Scholar 

  19. Halmos P.R.: Two subspaces. Trans. Am. Math. Soc. 144, 381–389 (1969)

    Article  MathSciNet  MATH  Google Scholar 

  20. Howland J.S.: On the Weinstein–Aronszajn formula. Arch. Rat. Mech. Anal. 39, 323–339 (1970)

    Article  MathSciNet  MATH  Google Scholar 

  21. Howland J.S.: Simple poles of operator-valued functions. J. Math. Anal. Appl. 36, 12–21 (1971)

    Article  MathSciNet  MATH  Google Scholar 

  22. Kalton N.: A note on pairs of projections. Bol. Soc. Mat. Mexicana (3) 3, 309–311 (1997)

    MathSciNet  MATH  Google Scholar 

  23. Kato T.: Notes on projections and perturbation theory, Technical Report No. 9, University of California, Berkeley, (unpublished) (1955)

  24. Kato T.: Wave operators and similarity for some non-selfadjoint operators. Math. Ann. 162, 258–279 (1966)

    Article  MathSciNet  MATH  Google Scholar 

  25. Kato T.: Perturbation Theory for Linear Operators, corr. printing of the 2nd ed. Springer, Berlin (1980)

    MATH  Google Scholar 

  26. Klaus M.: Some applications of the Birman–Schwinger principle. Helv. Phts. Acta. 55, 49–68 (1982)

    MathSciNet  Google Scholar 

  27. Klaus M., Simon B.: Coupling constant thresholds in nonrelativistic quantum mechanics. I. Short-range two-body case. Ann. Phys. 130, 251–281 (1980)

    Article  MathSciNet  MATH  Google Scholar 

  28. Konno R., Kuroda S.T.: On the finiteness of perturbed eigenvalues. J. Fac. Sci., Univ. Tokyo, Sec. I 13, 55–63 (1966)

    MathSciNet  MATH  Google Scholar 

  29. Kuroda S.T.: On a generalization of the Weinstein–Aronszajn formula and the infinite determinant. Sci. Papers Coll. Gen. Educ. Univ. Tokyo 11(1), 1–12 (1961)

    MathSciNet  MATH  Google Scholar 

  30. Latushkin Y., Sukhtayev A.: The algebraic multiplicity of eigenvalues and the Evans function revisited. Math. Model. Nat. Phenom. 5(4), 269–292 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  31. López-Gómez J., Mora-Corral C.: Algebraic Multiplicity of Eigenvalues of Linear Operators, Operator Theory: Advances and Applications, vol. 177. Birkhäuser, Basel (2007)

    MATH  Google Scholar 

  32. Magnus R.: On the multiplicity of an analytic operator-valued function. Math. Scand. 77, 108–118 (1995)

    MathSciNet  MATH  Google Scholar 

  33. Magnus R.: The spectrum and eigenspaces of a meromorphic operator-valued function. Proc. R. Soc. Edinb. 127, 1027–1051 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  34. Markus A.S.: Introduction to the Spectral Theory of Polynomial Operator Pencils, Transl. Math. Monographs, vol. 71. American Mathematical Society, Providence (1988)

    Google Scholar 

  35. Müller J., Strohmaier A.: The theory of Hahn meromorphic functions, a holomorphic Fredholm theorem, and its applications. Anal. PDE 7, 745–770 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  36. Newton R.G.: Bounds on the number of bound states for the Schrödinger equation in one and two dimensions. J. Oper. Theory 10, 119–125 (1983)

    MATH  Google Scholar 

  37. Pushnitski A.: The Birman–Schwinger principle on the essential spectrum. J. Funct. Anal. 261, 2053–2081 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  38. Rauch J.: Perturbation theory of eigenvalues and resonances of Schrödinger Hamiltonians. J. Funct. Anal. 35, 304–315 (1980)

    Article  MathSciNet  MATH  Google Scholar 

  39. Reed M., Simon B.: Methods of Modern Mathematical Physics. I: Functional Analysis, revised and enlarged edition. Academic Press, New York (1980)

    Google Scholar 

  40. Reed M., Simon B.: Methods of Modern Mathematical Physics. IV: Analysis of Operators. Academic Press, New York (1978)

    MATH  Google Scholar 

  41. Ribaric M., Vidav I.: Analytic properties of the inverse A(z)−1 of an analytic linear operator valued function A(z). Arch. Rat. Mech. Anal. 32, 298–310 (1969)

    Article  MathSciNet  MATH  Google Scholar 

  42. Schechter M.: Principles of Functional Analysis, Graduate Studies in Mathematics, vol. 36, 2nd edn. American Mathematical Society, Providence (2002)

    Google Scholar 

  43. Schwinger J.: On the bound states of a given potential. Proc. Natl. Acad. Sci. (USA) 47, 122–129 (1961)

    Article  MathSciNet  Google Scholar 

  44. Setô N.: Bargmann’s inequalities in spaces of arbitrary dimensions. Publ. RIMS, Kyoto Univ. 9, 429–461 (1974)

    Article  MATH  Google Scholar 

  45. Simon B.: Quantum Mechanics for Hamiltonians Defined as Quadratic Forms. Princeton University Press, Princeton (1971)

    MATH  Google Scholar 

  46. Simon B.: On the absorption of eigenvalues by continuous spectrum in regular perturbation problems. J. Funct. Anal. 25, 338–344 (1977)

    Article  MathSciNet  MATH  Google Scholar 

  47. Sinha K.: Index theorems in quantum mechanics. Math. Newsl. 19(1), 195–203 (2010)

    MathSciNet  MATH  Google Scholar 

  48. Steinberg S.: Meromorphic families of compact operators. Arch. Rat. Mech. Anal. 31, 372–379 (1968)

    Article  MathSciNet  MATH  Google Scholar 

  49. Weidmann J.: Linear Operators in Hilbert Spaces, Graduate Texts in Mathematics, vol. 68. Springer, New York (1980)

    Book  Google Scholar 

  50. Weinstein A., Stenger W.: Methods of Intermediate Problems for Eigenvalues. Academic Press, New York (1972)

    MATH  Google Scholar 

  51. Wolf F.: Analytic perturbation of operators in Banach spaces. Math. Ann. 124, 317–333 (1952)

    Article  MathSciNet  MATH  Google Scholar 

  52. Yafaev D.R.: Mathematical Scattering Theory, Transl. Math. Monographs, vol. 105. American Mathematical Society, Providence (1992)

    Google Scholar 

  53. Yafaev D.R.: Perturbation determinants, the spectral shift function, trace identities, and all that. Funct. Anal. Appl. 41, 217–236 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  54. Yafaev D.R.: Mathematical Scattering Theory. Analytic Theory, Math. Surveys and Monographs, vol. 158. American Mathematical Society, Providence (2010)

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, University of Missouri, Columbia, MO, 65211, USA

    Fritz Gesztesy

  2. Department of Mathematical Sciences, Norwegian University of Science and Technology, 7491, Trondheim, Norway

    Helge Holden

  3. Mathematics Department, The University of Tennessee at Chattanooga, 415 EMCS Building, Dept. 6956, 615 McCallie Ave, Chattanooga, TN, 37403, USA

    Roger Nichols

Authors
  1. Fritz Gesztesy
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Helge Holden
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Roger Nichols
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Fritz Gesztesy.

Additional information

Supported in part by the Research Council of Norway.

R. Nichols gratefully acknowledges support from an AMS–Simons Travel Grant.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Gesztesy, F., Holden, H. & Nichols, R. On Factorizations of Analytic Operator-Valued Functions and Eigenvalue Multiplicity Questions. Integr. Equ. Oper. Theory 82, 61–94 (2015). https://doi.org/10.1007/s00020-014-2200-7

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00020-014-2200-7

Mathematics Subject Classification

Keywords

Search

Navigation