Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

Root vectors for geometrically simple two-parameter eigenvalues


Authors: Paul Binding and Tomaz Kosir
Translated by:
Journal: Trans. Amer. Math. Soc. 356 (2004), 1705-1726
MSC (2000): Primary 35P10, 47A13; Secondary 35J55
DOI: https://doi.org/10.1090/S0002-9947-04-03542-1
Published electronically: January 6, 2004
MathSciNet review: 2031038
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: A class of two-parameter eigenvalue problems involving generally nonselfadjoint and unbounded operators is studied. A basis for the root subspace at a geometrically simple eigenvalue of Fredholm type is computed in terms of the underlying two-parameter system. Comparison with Faierman's work on two-parameter boundary value problems of Sturm-Liouville type is given as an application.


References [Enhancements On Off] (What's this?)

  • 1. F.V. Atkinson.
    Multiparameter Eigenvalue Problems.
    Academic Press, 1972. MR 56:9291
  • 2. P.A. Binding.
    Left Definite Multiparameter Eigenvalue Problems.
    Trans. Amer. Math. Soc., 272:475-486, 1982. MR 83h:47015
  • 3. P.A. Binding.
    Multiparameter Root Vectors.
    Proc. Edin. Math. Soc., 32:19-29, 1989. MR 90a:47048
  • 4. P.A. Binding and P.J. Browne.
    Asymptotics of Eigencurves for Second Order Ordinary Differential Equations I.
    J. Diff. Eq., 88:30-45, 1990. MR 92b:34101
  • 5. P.A. Binding and T. Kosir.
    Second Root Vectors for Multiparameter Eigenvalue Problems of Fredholm Type.
    Trans. Amer. Math. Soc., 348:229-249, 1996. MR 96f:35121
  • 6. P.A. Binding and K. Seddighi.
    Elliptic Multiparameter Eigenvalue Problems.
    Proc. Edin. Math. Soc., 30:215-228, 1987. MR 88g:47036
  • 7. G. Birkhoff.
    Lattice Theory, volume 25 of Amer. Math. Soc. Colloq. Publ., 3rd edition,
    Amer. Math. Soc., Providence, 1973. MR 37:2638
  • 8. M. Faierman.
    Two-parameter Eigenvalue Problems in Ordinary Differential Equations, volume 205 of Pitman Research Notes in Mathematics.
    Longman Scientific and Technical, Harlow, U.K., 1991. MR 93b:47095
  • 9. G.A. Gadzhiev.
    Introduction to Multiparameter Spectral Theory (in Russian),
    Azerbaijan State University, Baku, 1987.
  • 10. I.C. Gohberg, P. Lancaster, and L. Rodman.
    Invariant Subspaces of Matrices with Applications.
    Wiley-Interscience, 1986. MR 88a:15001
  • 11. L. Grunenfelder and T. Kosir.
    An Algebraic Approach to Multiparameter Eigenvalue Problems.
    Trans. Amer. Math. Soc., 348:2983-2998, 1996. MR 96j:47015
  • 12. L. Grunenfelder and T. Kosir.
    Coalgebras and Spectral Theory in One and Several Parameters.
    In the series Operator Theory: Adv. and Appl., 87: 177-192, Birkhäuser-Verlag, 1996. MR 97k:47019
  • 13. L. Grunenfelder and T. Kosir.
    Geometric Aspects of Multiparameter Spectral Theory,
    Trans. Amer. Math. Soc., 350:2525-2546, 1998. MR 98h:13032
  • 14. H.(G.A.) Isaev.
    Lectures on Multiparameter Spectral Theory.
    Dept. of Math. and Stats., University of Calgary, 1985.
  • 15. G.A. Isaev and A.S. Fainshtein.
    The Taylor Spectrum and Multiparameter Spectral Theory for Systems of Operators.
    Soviet Math. Dokl., 36:413-417, 1988. MR 88m:47028
  • 16. T. Kato.
    Perturbation Theory for Linear Operators, volume 132 of Grundlehren der math. Wiss.
    Springer-Verlag, second edition, 1984. MR 53:11389
  • 17. T. Kosir.
    Commuting Matrices and Multiparameter Eigenvalue Problems.
    PhD thesis, Dept. of Math. and Stats., University of Calgary, 1993.
  • 18. T. Kosir.
    On the Structure of Commutative Matrices.
    Lin. Alg. Appl., 187:163-182, 1993. MR 94i:15011
  • 19. T. Kosir.
    The Finite-Dimensional Multiparameter Spectral Theory: The Nonderogatory Case.
    Lin. Alg. Appl., 212/213:45-70, 1994. MR 95k:15020
  • 20. T. Kosir.
    On the Structure of Commutative Matrices II.
    Lin. Alg. Appl., 261:293-305, 1997. MR 98k:15018
  • 21. T. Kosir.
    Root Vectors for Geometrically Simple Multiparameter Eigenvalues,
    to appear in Int. Equat. Oper. Theory.
  • 22. J. Meixner, F. W. Schäfke, and G. Wolf.
    Mathieu Functions and Spheroidal Functions and Their Mathematical Foundations, volume 873 of Lect. Notes in Math.,
    Springer-Verlag, 1980. MR 83b:33013
  • 23. B. Plestenjak.
    A Numerical Algorithm for Computing a Basis for the Root Subspace at a Nonderogatory Eigenvalue of a Multiparameter System,
    Lin. Alg. Appl., 285:257-276, 1998. MR 2000a:65048
  • 24. B.D. Sleeman.
    Multiparameter Spectral Theory in Hilbert Space, volume 22 of Pitman Research Notes in Mathematics.
    Pitman Publ. Ltd., London U.K., Belmont U.S.A., 1978. MR 81h:47004
  • 25. A.E. Taylor and D.C. Lay.
    Introduction to Functional Analysis.
    Wiley, New York, second edition, 1980. MR 81b:46001
  • 26. H. Volkmer.
    Multiparameter Eigenvalue Problems and Expansion Theorems, volume 1356 of Lecture Notes in Mathematics.
    Springer-Verlag, Berlin, New York, 1988. MR 90d:47021

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (2000): 35P10, 47A13, 35J55

Retrieve articles in all journals with MSC (2000): 35P10, 47A13, 35J55


Additional Information

Paul Binding
Affiliation: Department of Mathematics and Statistics, University of Calgary, Calgary, Alberta, Canada T2N 1N4
Email: binding@ucalgary.ca

Tomaz Kosir
Affiliation: Department of Mathematics, University of Ljubljana, Jadranska 19, 1000 Ljubljana, Slovenia
Email: tomaz.kosir@fmf.uni-lj.si

DOI: https://doi.org/10.1090/S0002-9947-04-03542-1
Received by editor(s): June 15, 2001
Published electronically: January 6, 2004
Additional Notes: The first author’s research was supported by NSERC of Canada
The second author’s research was supported by the Ministry of Science and Technology of Slovenia
Article copyright: © Copyright 2004 American Mathematical Society

American Mathematical Society