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On the spectrum and numerical range of tridiagonal random operators

Hagger, R. (2016) On the spectrum and numerical range of tridiagonal random operators. Journal of Spectral Theory, 6 (2). pp. 215-266. ISSN 1664-039X

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To link to this item DOI: 10.4171/JST/124

Abstract/Summary

In this paper we derive an explicit formula for the numerical range of (non-self-adjoint) tridiagonal random operators. As a corollary we obtain that the numerical range of such an operator is always the convex hull of its spectrum, this (surprisingly) holding whether or not the random operator is normal. Furthermore, we introduce a method to compute numerical ranges of (not necessarily random) tridiagonal operators that is based on the Schur test. In a somewhat combinatorial approach we use this method to compute the numerical range of the square of the (generalized) Feinberg-Zee random hopping matrix to obtain an improved upper bound to the spectrum. In particular, we show that the spectrum of the Feinberg-Zee random hopping matrix is not convex.

Item Type:Article
Refereed:Yes
Divisions:Faculty of Science > School of Mathematical, Physical and Computational Sciences > Department of Mathematics and Statistics
ID Code:84008
Publisher:European Mathematical Society

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