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Darboux transformation of generalized Jacobi matrices

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Abstract

Let $\mathfrak{J}$ be a monic generalized Jacobi matrix, i.e. a three-diagonal block matrix of special form, introduced by M.~Derevyagin and V.~Derkach in 2004. We find conditions for a monic generalized Jacobi matrix $\mathfrak{J}$ to admit a factorization $\mathfrak{J}=\mathfrak{LU}$ with $\mathfrak{L}$ and $\mathfrak{U}$ being lower and upper triangular two-diagonal block matrices of special form. In this case the Darboux transformation of $\mathfrak{J}$ defined by $\mathfrak{J}^{(p)}=\mathfrak{UL}$ is shown to be also a monic generalized Jacobi matrix. Analogues of Christoffel formulas for polynomials of the first and the second kind, corresponding to the Darboux transformation $\mathfrak{J}^{(p)}$ are found.

Key words: Darboux transformation, indefinite inner product, m-function, monic generalized Jacobi matrix, triangular factorization.


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Article Information

TitleDarboux transformation of generalized Jacobi matrices
SourceMethods Funct. Anal. Topology, Vol. 20 (2014), no. 4, 301-320
MathSciNet MR3309669
zbMATH 1324.47058
MilestonesReceived 11/03/2014
CopyrightThe Author(s) 2014 (CC BY-SA)

Authors Information

Ivan Kovalyov
Department of Mathematics, Donetsk National University, 24 Universytetska, Donetsk, 83055, Ukraine


Citation Example

Ivan Kovalyov, Darboux transformation of generalized Jacobi matrices, Methods Funct. Anal. Topology 20 (2014), no. 4, 301-320.


BibTex

@article {MFAT741,
    AUTHOR = {Kovalyov, Ivan},
     TITLE = {Darboux transformation of generalized  Jacobi matrices},
   JOURNAL = {Methods Funct. Anal. Topology},
  FJOURNAL = {Methods of Functional Analysis and Topology},
    VOLUME = {20},
      YEAR = {2014},
    NUMBER = {4},
     PAGES = {301-320},
      ISSN = {1029-3531},
  MRNUMBER = {MR3309669},
 ZBLNUMBER = {1324.47058},
       URL = {http://mfat.imath.kiev.ua/article/?id=741},
}


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