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# Crouzeix's conjecture holds for tridiagonal 3x3 matrices with elliptic numerical range centered at an eigenvalue

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Publication Details

List of Authors: Christer Glader, Mikael Kurula, Mikael Lindström

Publisher: arXiv.org

Publication year: 2017

Abstract

M. Crouzeix formulated the following conjecture in (Integral Equations

Operator Theory 48, 2004, 461--477): For every square matrix A and every

polynomial p,

of A. We show that the conjecture holds in its strong, completely bounded

form, i.e., where p above is allowed to be any matrix-valued polynomial, for

all tridiagonal 3×3 matrices with constant main diagonal: ⎡⎣⎢ac10b1ac20b2a⎤⎦⎥,a,bk,ck∈C, or equivalently, for all complex 3×3 matrices

with elliptic numerical range and one eigenvalue at the center of the ellipse.

We also extend the main result of D. Choi in (Linear Algebra Appl. 438,

3247--3257) slightly.