Abstract
For the periodic matrix-valued Jacobi operator J we obtain the estimate of the Lebesgue measure of the spectrum \(\mathrm{mes}(\sigma (J))\leqslant 4\min _{n}\mathop{ \mathrm{Tr}}\nolimits (a_{n}a_{n}^{{\ast}})^{\frac{1} {2} }\), where a n are off-diagonal elements of J.
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Acknowledgements
I would like to express gratitude to Prof. E. Korotyaev for stimulating discussions and helpful remarks, to Prof. B. Simon for useful comments and for the reference to the paper [8], and to Prof. Michael J. Gruber, who has advised me to use \(\mathop{\mathrm{Tr}}\nolimits (a_{n}a_{n}^{{\ast}})^{\frac{1} {2} }\) instead of \(\|a_{n}\|\mathop{ \mathrm{rank}}a_{n}\) in (2) (which was in the first version of this paper 2010).
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Kutsenko, A.A. (2015). Sharp Spectral Estimates for Periodic Matrix-Valued Jacobi Operators. In: Mugnolo, D. (eds) Mathematical Technology of Networks. Springer Proceedings in Mathematics & Statistics, vol 128. Springer, Cham. https://doi.org/10.1007/978-3-319-16619-3_9
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DOI: https://doi.org/10.1007/978-3-319-16619-3_9
Publisher Name: Springer, Cham
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