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# Crouzeix's Conjecture Holds for Tridiagonal 3 x 3 Matrices with Elliptic Numerical Range Centered at an Eigenvalue

A1 Journal article (refereed)

Internal Authors/Editors

Publication Details

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

Publisher: Society for Industrial and Applied Mathematics

Publication year: 2018

Journal: SIAM Journal on Matrix Analysis and Applications

Volume number: 39

Issue number: 1

Start page: 346

End page: 364

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.

Crouzeix

stated the following conjecture in [Integral Equations Operator Theory,

48 (2004), pp. 461-477]: For every n x n matrix A and every polynomial

p, |p(A)| le 2 max_z in W(A)|p(z)|, where W(A) is the numerical range

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 x 3 matrices with constant main

diagonal, [smallmatrix a b_1 0, c_1 a b_2, 0 c_2 a], a,b_k,c_k in C, or

equivalently, for all complex 3 x 3 matrices with elliptic numerical

range and one eigenvalue at the center of the ellipse. We also extend

the main result of Choi in [Linear Algebra Appl., 438 (2013), pp.

3247-3257] slightly.

Read More: https://epubs.siam.org/doi/abs/10.1137/17M1110663

Crouzeix

stated the following conjecture in [Integral Equations Operator Theory,

48 (2004), pp. 461-477]: For every n x n matrix A and every polynomial

p, |p(A)| le 2 max_z in W(A)|p(z)|, where W(A) is the numerical range

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 x 3 matrices with constant main

diagonal, [smallmatrix a b_1 0, c_1 a b_2, 0 c_2 a], a,b_k,c_k in C, or

equivalently, for all complex 3 x 3 matrices with elliptic numerical

range and one eigenvalue at the center of the ellipse. We also extend

the main result of Choi in [Linear Algebra Appl., 438 (2013), pp.

3247-3257] slightly.

Read More: https://epubs.siam.org/doi/abs/10.1137/17M1110663

Keywords

3 x 3 matrix, Crouzeix's conjecture, elliptic numerical range