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Faber Polynomials and Spectrum Localisation

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Abstract

Let \(K\) be a compact connected subset of the complex plane, of non-void interior, and whose complement in the extended complex plane is connected. Denote by \(F_n\) the \(n\)th Faber polynomial associated with \(K\). The aim of this paper is to find suitable Banach spaces of complex sequences, \(\mathcal{R },\) such that statements of the following type hold true: if \(T\) is a bounded linear operator acting on the Banach space \(\mathcal{X }\) such that \(( \langle F_n(T)x,x^*\rangle )_{n\ge 0} \in \mathcal{R }\) for each pair \((x,x^{*}) \in \mathcal{X }\times \mathcal{X }^{*}\), then the spectrum of \(T\) is included in the interior of \(K\). Generalisations of some results due to W. Mlak, N. Nikolski and J. van Neerven are, thus, obtained and several illustrative examples are given. An interesting feature of these generalisations is the influence of the geometry of \(K\) and the regularity of its boundary.

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Acknowledgments

I would like to express my sincere gratitude to my advisor Catalin Badea for proposing the topic of this paper and for his guidance and support. I would also like to thank the two referees for their constructive remarks, critical reading and suggestions for improvements.

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  1. Laboratoire Paul Painlevé, UMR CNRS 8524, Bât. M2, Université Lille 1, 59655 , Villeneuve d’Ascq Cedex, France

    Oscar Devys

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  1. Oscar Devys
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Correspondence to Oscar Devys.

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Communicated by Edward B. Saff.

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Devys, O. Faber Polynomials and Spectrum Localisation. Comput. Methods Funct. Theory 13, 107–131 (2013). https://doi.org/10.1007/s40315-013-0010-6

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