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# The direct and inverse spectral problems for (2N+1)-diagonal complex transposition-antisymmetric matrices

### Abstract

We consider a difference equation associated with a semi-infinite complex $(2N+1)$-diagonal transposition-antisymmetric matrix $J=(g_{k,l})_{k,l=0}^\infty$ with $g_{k,k+N} \not=0$, $k=0,1,2,\ldots ,$ ($g_{k,l}=-g_{l,k}$): $\sum_{j=-N}^N g_{k,k+j} y_{k+j} = \lambda^N y_k,\ k=0,1,2,\ldots ,$ where $y=(y_0,y_1,y_2,\ldots )$ is an unknown vector, $\lambda$ is a complex parameter, $g_{k,l}$ and $y_l$ with negative indices are equal to zero, $N\in\mathbb N$. We introduce a notion of the spectral function for this difference equation. We state and solve the direct and inverse problems for this equation.

### Article Information

 Title The direct and inverse spectral problems for (2N+1)-diagonal complex transposition-antisymmetric matrices Source Methods Funct. Anal. Topology, Vol. 14 (2008), no. 2, 124-131 MathSciNet MR2432761 Copyright The Author(s) 2008 (CC BY-SA)

### Authors Information

S. M. Zagorodnyuk
School of Mathematics and Mechanics, Karazin Kharkiv National University, 4 Svobody sq., Kharkiv, 61077, Ukraine

### Citation Example

S. M. Zagorodnyuk, The direct and inverse spectral problems for (2N+1)-diagonal complex transposition-antisymmetric matrices, Methods Funct. Anal. Topology 14 (2008), no. 2, 124-131.

### BibTex

@article {MFAT450,
AUTHOR = {Zagorodnyuk, S. M.},
TITLE = {The direct and inverse spectral problems for (2N+1)-diagonal complex transposition-antisymmetric matrices},
JOURNAL = {Methods Funct. Anal. Topology},
FJOURNAL = {Methods of Functional Analysis and Topology},
VOLUME = {14},
YEAR = {2008},
NUMBER = {2},
PAGES = {124-131},
ISSN = {1029-3531},
URL = {http://mfat.imath.kiev.ua/article/?id=450},
}