On the spectrum and numerical range of tridiagonal random operators | Zendy

Raffael Hagger | Zendy

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On the spectrum and numerical range of tridiagonal random operators

Author(s) -

Raffael Hagger

Publication year - 2016

Publication title -

journal of spectral theory

Language(s) - English

Resource type - Journals

SCImago Journal Rank - 1.063

H-Index - 19

eISSN - 1664-0403

pISSN - 1664-039X

DOI - 10.4171/jst/124

Subject(s) - numerical range , tridiagonal matrix , mathematics , spectrum (functional analysis) , operator (biology) , random matrix , convex hull , range (aeronautics) , corollary , matrix (chemical analysis) , regular polygon , numerical analysis , mathematical analysis , pure mathematics , eigenvalues and eigenvectors , geometry , quantum mechanics , physics , biochemistry , chemistry , materials science , repressor , transcription factor , composite material , gene

In this paper we derive an explicit formula for the numerical range of (non-selfadjoint) 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 approachwe use thismethod 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. © European Mathematical Society

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