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Darboux transformation of generalized Jacobi matrices


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

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

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


@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 = {},

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