Pure and Applied Mathematics Journal

Special Issue

Spectral Theory of Multiparameter Operator Pencils and Its Applications

  • Submission Deadline: May 10, 2015
  • Status: Submission Closed
  • Lead Guest Editor: Rakhshanda Dzhabarzadeh
About This Special Issue
Spectral theory of operators is one of the important directions of functional analysis. The spectral theory of operator pencils is raised as the result of study of the problems of ordinary differential equations with the boundary conditions. But the multiparameter spectral theory is raised in the result of the study of the problems of the partial differential equations and the equations of the mathematical physics.The physical sciences open more and more challenges for mathematicians. In particular, the research of the problems associated with the physical processes and, consequently, the study of partial differential equations and mathematical physics equations, required a new approach. The method of separation of variables in many cases turned out to be the only acceptable, since it reduces finding a solution to a complex equation with many variables to find a solution to a system of ordinary differential equations, which are much easier to study. For example, a multivariable problems cause problems in quantum mechanics, diffraction theory, the theory of elastic shells, nuclear reactor calculations , stochastic diffusion processes, Brownian motion, boundary value problems for equations of elliptic-parabolic type, the Cauchy problem for ultraparabolic equations and etc.

Despite the urgency and prescription studies, spectral theory of multiparameter systems studied was not enough. The available results in this area until recently only dealt with seltadjoint multiparameter systems. Original research papers in this area will help further explored this actual direction of functional analysis.

It is expected to pay more attention to the study of the particular case of nonlinear multiparameter systems of operators in the Hilbert space, namely, to the operator pencils and nonlinear algebraic systems with many variables.

Aims and Scope:

1. Spectral problem of multiparameter system
2. Operator pencils in Hilbert space.
3. Problem of completeness of eigen and associated vectors of multiparameter systems of operator pencils in Hilbert spaces.
4. Nonlinear algebraic equations with many variables.
5. Bases of eigen and associated vectors of multiparameter system.
6. Application of results of multiparameter spectral theory.
7. Fundamental notions of spectral theory of operators.
Lead Guest Editor
  • Rakhshanda Dzhabarzadeh

    Department of Functional Analysis, Institute of Mathematics and Mechanics, Nathional Academy of Sciencis of Azerbaijan, Baku, Azerbaijan

Guest Editors
  • Ilgar Jafarov

    Institute of Mathematics and Mechanics of NAS of Azerbaijan, Baku, Azerbaijan

Published Articles
  • Research Methods of Multiparameter System in Hilbert Spaces

    Rakhshanda Dzhabarzadeh

    Issue: Volume 4, Issue 4-1, August 2015
    Pages: 38-44
    Received: May 09, 2015
    Accepted: May 19, 2015
    Published: Aug. 24, 2015
    DOI: 10.11648/j.pamj.s.2015040401.18
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    Abstract: The work is devoted to the presentation of the methods, available in the literature, of the study of multiparameter spectral problems in Hilbert space. In particular, the method of Atkinson and his followers for a purely self-adjoint multiparameter systems and methods proposed by the author for the study, in general, non- selfadjoint multiparameter... Show More
  • Spectral Problems of Two-Parameter System of Operators

    Rakhshanda Dzhabarzadeh

    Issue: Volume 4, Issue 4-1, August 2015
    Pages: 33-37
    Received: May 04, 2015
    Accepted: May 19, 2015
    Published: Aug. 21, 2015
    DOI: 10.11648/j.pamj.s.2015040401.17
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    Abstract: The author has proved the existence of the multiple basis of the eigen and associated vectors of the two parameter system of operators in Hilbert spaces. The proof essentially uses the theorem of the existence of multiple basis of operator bundles and the notion of the abstract analog of resultant of two operator pencils, acting, generally speaking... Show More
  • On Multiple Bases of Eigen and Associated Vectors of Operator Pencils in the Hilbert Spaces

    Rakhshanda Dzhabarzadeh

    Issue: Volume 4, Issue 4-1, August 2015
    Pages: 27-32
    Received: May 04, 2015
    Accepted: May 19, 2015
    Published: Aug. 21, 2015
    DOI: 10.11648/j.pamj.s.2015040401.16
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    Abstract: It is proved the theorem about of multiple basis of eigen and associated vectors of the operator pencil, non-linear depending on parameter in the Hilbert space. This work is the generalization of existing results on the multiple completeness of the eigen and associated vectors of polynomial pencils, rationally depending on parameters. At the proof ... Show More
  • Spectral Theory of Operator Pencils in the Hilbert Spaces

    Rakhshanda Dzhabarzadeh

    Issue: Volume 4, Issue 4-1, August 2015
    Pages: 22-26
    Received: Apr. 29, 2015
    Accepted: May 14, 2015
    Published: Aug. 21, 2015
    DOI: 10.11648/j.pamj.s.2015040401.15
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    Abstract: The theorem on possibility of multiple summation of the series on eigen and associated vectors of the operator pencil in the Hilbert space is proved. Research of multiple completeness and multiple expansions of eigen and associated vectors of such operator pencils are closely connected with the research of differential operator equation with the bo... Show More
  • On Existence of Eigen Values of Several Operator Bundles with Two Parameters

    Makhmudova Malaka Gasan , Sultanova Elnara Bayram

    Issue: Volume 4, Issue 4-1, August 2015
    Pages: 16-21
    Received: Apr. 19, 2015
    Accepted: May 14, 2015
    Published: Aug. 21, 2015
    DOI: 10.11648/j.pamj.s.2015040401.14
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    Abstract: For the several operator bundles with two parameters when the number of equation is greater than the number of parameters in the Hilbert spaces is given the criterion of existence of the common point of spectra. In the special case the common point of spectra is the common eigen value. In the proof of the theorem the authors use the results of the ... Show More
  • Criterion of Existence of Eigen Values of Linear Multiparameter Systems

    Rakhshanda Dzhabarzadeh , Elnara Sultanova

    Issue: Volume 4, Issue 4-1, August 2015
    Pages: 11-15
    Received: Mar. 23, 2015
    Accepted: Apr. 10, 2015
    Published: May 12, 2015
    DOI: 10.11648/j.pamj.s.2015040401.13
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    Abstract: It is considered the linear multiparameter system of operators when the number of equations may be more than the number of parameters. For such multiparameter systems the authors have proved the criterion of existence of eigen values. Under certain conditions, the authors a have proved that all components of the eigen values of the considered multi... Show More
  • Multiparameter Operator Systems with Three Parameters

    Rakhshanda Dzhabarzadeh , Kamilla Alimardanova

    Issue: Volume 4, Issue 4-1, August 2015
    Pages: 5-10
    Received: Feb. 25, 2015
    Accepted: Feb. 27, 2015
    Published: May 12, 2015
    DOI: 10.11648/j.pamj.s.2015040401.12
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    Abstract: For the multiparameter system of operators in three parameters the conditions of the existence of multiple basis of eigen and associated vectors in finite dimensional space is proved. The proof of this fact uses essentially the notion of the Resultant of two operator pencils, acting in, generally speaking, in different Hilbert spaces and the criter... Show More
  • Multiparameter System of Operators with Two Parameters in Finite Dimensional Spaces

    Rakhshanda Dzhabarzadeh , Afet Jabrailova

    Issue: Volume 4, Issue 4-1, August 2015
    Pages: 1-4
    Received: Feb. 25, 2015
    Accepted: Feb. 27, 2015
    Published: May 12, 2015
    DOI: 10.11648/j.pamj.s.2015040401.11
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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... Show More