On the Spectrum and Numerical Range of Tridiagonal Random Operators

TL;DR

This paper provides an explicit formula for the numerical range of non-self-adjoint tridiagonal random operators, showing it is always the convex hull of the spectrum, and introduces a new method for computing numerical ranges.

Contribution

It derives a formula for the numerical range of tridiagonal random operators and presents a novel method based on the Schur test for computing numerical ranges.

Findings

Numerical range equals the convex hull of the spectrum for these operators.

The spectrum of the Feinberg-Zee random hopping matrix is not convex.

An improved upper bound for the spectrum was obtained using the new method.

Abstract

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.