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# Darboux transformation of generalized Jacobi matrices

### 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.

### Article Information

 Title Darboux transformation of generalized Jacobi matrices Source Methods Funct. Anal. Topology, Vol. 20 (2014), no. 4, 301-320 MathSciNet MR3309669 zbMATH 1324.47058 Milestones Received 11/03/2014 Copyright The 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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