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Coalgebras and Spectral Theory in One and Several Parameters

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Recent Developments in Operator Theory and Its Applications

Part of the book series: Operator Theory Advances and Applications ((OT,volume 87))

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Abstract

The coalgebraic versions of the primary decomposition theorem for a single linear map and for several commuting linear maps are proved. They lead to a description of the primary decomposition for multiparameter eigenvalue problems in terms of the underlying multiparameter system. Also the coalgebraic version of the primary decomposition theorem for a monic matrix polynomial is discussed.

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References

  1. E. Abe. Hopf Algebras. Cambridge Univ. Press, 1980.

    MATH  Google Scholar 

  2. F.V. Atkinson. Multiparameter Spectral Theory. Bull. Amer. Math. Soc., 74:1–27, 1968.

    Article  MathSciNet  MATH  Google Scholar 

  3. F.V. Atkinson. Multiparameter Eigenvalue Problems. Academic Press, 1972.

    MATH  Google Scholar 

  4. P.A. Binding. Multiparameter Root Vectors. Proc. Edin. Math. Soc., 32:19–29, 1989.

    Article  MathSciNet  MATH  Google Scholar 

  5. C. Davis. Representing a Commuting Pair by Tensor Products. Lin. Alg. Appl., 3:355–357, 1970.

    Article  MATH  Google Scholar 

  6. 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.

    MATH  Google Scholar 

  7. C.K. Fong and A.R. Sourour. Renorming, Similarity and Numerical Ranges. J. London Math. Soc., (2)18:511–518, 1978.

    Article  MathSciNet  MATH  Google Scholar 

  8. G.A. Gadzhiev. On a Multitime Equation and its Reduction to a Multiparameter Spectral Problem. Soviet Math. Dokl, 32:710–713, 1985.

    MATH  Google Scholar 

  9. G.A. Gadzhiev. Introduction to Multiparameter Spectral Theory (in Russian). Azerbaijan State University, Baku, 1987.

    MATH  Google Scholar 

  10. I. Gohberg, P. Lancaster, and L. Rodman. Matrix Polynomials. Academic Press, 1982.

    MATH  Google Scholar 

  11. L. Grunenfelder. Hopf-Algebren und Coradical. Math. Z., 116:166–182, 1970.

    Article  MathSciNet  Google Scholar 

  12. L. Grunenfelder and T. Košir. Representation of Commuting Maps by Tensor Products, to appear in Lin. Alg. Appl.

    Google Scholar 

  13. L. Grunenfelder and T. Košir. An Algebraic Approach to Multiparameter Eigenvalue Problems, submitted.

    Google Scholar 

  14. L. Grunenfelder and T. Košir. Koszul Cohomology for Finite Families of Comodule Maps and Applications, submitted.

    Google Scholar 

  15. L. Grunenfelder and M. Omladič. Linearly Recursive Sequences and Operator Polynomials. Lin. Alg. Appl, 182:127–145, 1993.

    Article  MATH  Google Scholar 

  16. L. Grunenfelder and R. Paré. Families Parametrized by Coalgebras. J. Algebra, 107:316–375, 1987.

    Article  MathSciNet  MATH  Google Scholar 

  17. K. Hoffman and R. Kunze. Linear Algebra. Prentice Hall, second edition, 1971.

    MATH  Google Scholar 

  18. G.A. Isaev. On Root Elements of Multiparameter Spectral Problems. Soviet. Math. Dokl., 21:127–130, 1980.

    MATH  Google Scholar 

  19. H.(G.A.) Isaev. Lectures on Multiparameter Spectral Theory. Dept. of Math, and Stats., University of Calgary, 1985.

    Google Scholar 

  20. T. Košir. Commuting Matrices and Multiparameter Eigenvalue Problems. Ph.D. thesis, Dept. of Math, and Stats., University of Calgary, 1993.

    Google Scholar 

  21. T. Košir. The Finite-dimensional Multiparameter Spectral Theory: The Nonderogatory Case. Lin. Alg. Appl, 212/213:45–70, 1994.

    Article  Google Scholar 

  22. S. Montgomery. Hopf Algebras and Their Actions on Rings. CBMS Reg. Conf. Ser. in Math. 82, AMS, 1993.

    MATH  Google Scholar 

  23. B. Peterson and E. Taft. The Hopf Algebra of Linearly Recursive Sequences. Aequationes Math., 20:1–17, 1980.

    Article  MathSciNet  MATH  Google Scholar 

  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.

    MATH  Google Scholar 

  25. M.E. Sweedler. Hopf Algebras. Benjamin, New York, 1969.

    Google Scholar 

  26. H. Volkmer. Multiparameter Eigenvalue Problems and Expansion Theorems, volume 1356 of Lecture Notes in Mathematics. Springer-Verlag, Berlin, New York, 1988.

    MATH  Google Scholar 

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Author information

Authors and Affiliations

  1. Department of Mathematics, Statistics and Computing Science, Dalhousie University, Halifax, Nova Scotia, B3H 3J5, Canada

    Luzius Grunenfelder & Tomaž Košir

Authors
  1. Luzius Grunenfelder
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  2. Tomaž Košir
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Editor information

Editors and Affiliations

  1. School of Mathematical Sciences, Tel Aviv University, 69978, Tel Aviv, Israel

    I. Gohberg

  2. Department of Mathematics and Statistics, University of Calgary, Calgary, Alberta, T2N 1NA, Canada

    P. Lancaster

  3. Department of Applied Mathematics and Institute of Industrial Mathematical Sciences, University of Manitoba, Winnipeg, Manitoba, R3T 2N2, Canada

    P. N. Shivakumar

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© 1996 Birkhäuser Verlag Basel/Switzerland

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Grunenfelder, L., Košir, T. (1996). Coalgebras and Spectral Theory in One and Several Parameters. In: Gohberg, I., Lancaster, P., Shivakumar, P.N. (eds) Recent Developments in Operator Theory and Its Applications. Operator Theory Advances and Applications, vol 87. Birkhäuser Basel. https://doi.org/10.1007/978-3-0348-9035-9_8

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  • DOI: https://doi.org/10.1007/978-3-0348-9035-9_8

  • Publisher Name: Birkhäuser Basel

  • Print ISBN: 978-3-0348-9878-2

  • Online ISBN: 978-3-0348-9035-9

  • eBook Packages: Springer Book Archive

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