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Multiparameter System of Operators with Two Parameters in Finite Dimensional Spaces

Received: 25 February 2015     Accepted: 27 February 2015     Published: 12 May 2015
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Abstract

The authors have proved the existence of the multiple basis on the eigen and associated elements of the two parameter system of operators in finite dimensional spaces. The proof uses the notion of the abstract analog of resultant of two operator pencils, acting, generally speaking, in different Hilbert spaces. In this paper necessary and sufficient conditions of the existence of multiple completeness of the eigen and associated vectors of two parameter system of operators in finite dimensional Hilbert space is given.

Published in Pure and Applied Mathematics Journal (Volume 4, Issue 4-1)

This article belongs to the Special Issue Spectral Theory of Multiparameter Operator Pencils and Its Applications

DOI 10.11648/j.pamj.s.2015040401.11
Page(s) 1-4
Creative Commons

This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited.

Copyright

Copyright © The Author(s), 2015. Published by Science Publishing Group

Keywords

Multiparameter, Spectrum, Operator, Space, Eigenvector

References
[1] Atkinson F.V. Multiparameter spectral theory. Bull.Amer. Math.Soc.1968, 74, 1-27.
[2] Browne P.J. Multiparameter spectral theory. Indiana Univ. Math. J,24, 3, 1974
[3] Sleeman B.D. Multiparameter spectral theory in Hilbert space. Pitnam Press, London, 1978, p.118.
[4] Dzhabarzadeh R.M. Spectral theory of two parameter system in finite-dimensional space. Transactions of NAS Azerbaijan, v. 3-4 1998, p.12-18
[5] Dzhabarzadeh R.M. Spectral theory of multiparameter system of operators in Hilbert space, Transactions of NAS of Azerbaijan, 1-2, 1999, 33-41
[6] Balinskii A.I (Балинский) Generation of notions of Bezutiant and Resultant DAN of Ukr. SSR, ser.ph.-math and tech. of sciences, 1980, 2. (in Russian).
[7] Khayniq (Хайниг Г). Abstract analog of Resultant for two polynomial bundles Functional analyses and its applications, 1977, 2 , no. 3, p.94-95
[8] Dzhabarzadeh R.M. On solutions of nonlinear algebraic systems with two variables. Pure and Applied Mathematics Journal, vol. 2, No. 1, pp. 31-37, 2013
[9] Keldish M .V. On completeness of eigen functions of some classes of linear nonselfadjoint operators .Successes of Mathematical Sciences (УМН), 1971, v.27, issue.4, pp.15-47 (in Russian)
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  • APA Style

    Rakhshanda Dzhabarzadeh, Afet Jabrailova. (2015). Multiparameter System of Operators with Two Parameters in Finite Dimensional Spaces. Pure and Applied Mathematics Journal, 4(4-1), 1-4. https://doi.org/10.11648/j.pamj.s.2015040401.11

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    ACS Style

    Rakhshanda Dzhabarzadeh; Afet Jabrailova. Multiparameter System of Operators with Two Parameters in Finite Dimensional Spaces. Pure Appl. Math. J. 2015, 4(4-1), 1-4. doi: 10.11648/j.pamj.s.2015040401.11

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    AMA Style

    Rakhshanda Dzhabarzadeh, Afet Jabrailova. Multiparameter System of Operators with Two Parameters in Finite Dimensional Spaces. Pure Appl Math J. 2015;4(4-1):1-4. doi: 10.11648/j.pamj.s.2015040401.11

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  • @article{10.11648/j.pamj.s.2015040401.11,
      author = {Rakhshanda Dzhabarzadeh and Afet Jabrailova},
      title = {Multiparameter System of Operators with Two Parameters in Finite Dimensional Spaces},
      journal = {Pure and Applied Mathematics Journal},
      volume = {4},
      number = {4-1},
      pages = {1-4},
      doi = {10.11648/j.pamj.s.2015040401.11},
      url = {https://doi.org/10.11648/j.pamj.s.2015040401.11},
      eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.pamj.s.2015040401.11},
      abstract = {The authors have proved the existence of the multiple basis on the eigen and associated elements of the two parameter system of operators in finite dimensional spaces. The proof uses the notion of the abstract analog of resultant of two operator pencils, acting, generally speaking, in different Hilbert spaces. In this paper necessary and sufficient conditions of the existence of multiple completeness of the eigen and associated vectors of two parameter system of operators in finite dimensional Hilbert space is given.},
     year = {2015}
    }
    

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    T1  - Multiparameter System of Operators with Two Parameters in Finite Dimensional Spaces
    AU  - Rakhshanda Dzhabarzadeh
    AU  - Afet Jabrailova
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    N1  - https://doi.org/10.11648/j.pamj.s.2015040401.11
    DO  - 10.11648/j.pamj.s.2015040401.11
    T2  - Pure and Applied Mathematics Journal
    JF  - Pure and Applied Mathematics Journal
    JO  - Pure and Applied Mathematics Journal
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    PB  - Science Publishing Group
    SN  - 2326-9812
    UR  - https://doi.org/10.11648/j.pamj.s.2015040401.11
    AB  - The authors have proved the existence of the multiple basis on the eigen and associated elements of the two parameter system of operators in finite dimensional spaces. The proof uses the notion of the abstract analog of resultant of two operator pencils, acting, generally speaking, in different Hilbert spaces. In this paper necessary and sufficient conditions of the existence of multiple completeness of the eigen and associated vectors of two parameter system of operators in finite dimensional Hilbert space is given.
    VL  - 4
    IS  - 4-1
    ER  - 

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Author Information
  • Department of functional analysis, Institute of Mathematics and Mechanics of NAS of Azerbaijan, Baku

  • Department of functional analysis, Institute of Mathematics and Mechanics of NAS of Azerbaijan, Baku

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