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On a Family of Laurent Polynomials Generated by \(\varvec{2\times 2}\) Matrices

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Abstract

To \(2\times 2\) matrix G with complex entries, the sequence of Laurent polynomial \(L_n(z,G)={{\mathrm{tr}}}\left( G\left[ \begin{matrix} z&{}0\\ 0&{}z^{-1} \end{matrix} \right] G^{*}\right) ^n\) is related. It turns out that for each n, the family \(\big \{L_n(z,G)\big \}_G\), where G runs over the set of all \(2\times 2\) matrices, is a three-parametric family. A natural parametrization of this family is found. The polynomial \(L_n(z,G)\) is expressed in terms of these parameters and the Chebyshev polynomial \(T_n\). The zero set of the polynomial \(L_n(z,G)\) is described.

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Notes

  1. There are \(2^n\) such combinations.

  2. Here \(\ln z=0\) for \(z=1\) and \(\sqrt{z^2-1}>0\) for \(z\in (1,+\infty )\).

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  1. Department of Mathematics, The Weizmann Institute, 76100, Rehovot, Israel

    Victor Katsnelson

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  1. Victor Katsnelson
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Correspondence to Victor Katsnelson.

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Communicated by Bernd Kirstein.

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Katsnelson, V. On a Family of Laurent Polynomials Generated by \(\varvec{2\times 2}\) Matrices. Complex Anal. Oper. Theory 11, 857–873 (2017). https://doi.org/10.1007/s11785-016-0568-x

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  • DOI: https://doi.org/10.1007/s11785-016-0568-x

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