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Faber Polynomials and Spectrum Localisation

Faber Polynomials and Spectrum Localisation 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. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Computational Methods and Function Theory Springer Journals

Faber Polynomials and Spectrum Localisation

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References (26)

Publisher
Springer Journals
Copyright
Copyright © 2013 by Springer-Verlag Berlin Heidelberg
Subject
Mathematics; Analysis; Computational Mathematics and Numerical Analysis; Functions of a Complex Variable
ISSN
1617-9447
eISSN
2195-3724
DOI
10.1007/s40315-013-0010-6
Publisher site
See Article on Publisher Site

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.

Journal

Computational Methods and Function Theory – Springer Journals

Published: Mar 6, 2013

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