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

Christer Glader; Mikael Kurula; Mikael Lindström

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

Christer Glader, Mikael Kurula, Mikael Lindström

Mathematics

Research output: Book/Journal/Report › Commissioned report › Professional

Abstract

M. Crouzeix formulated the following conjecture in (Integral EquationsOperator Theory 48, 2004, 461--477): For every square matrix A and everypolynomial p,

∥p(A)∥≤2maxz∈W(A)|p(z)|, where W(A) is the numerical rangeof A. We show that the conjecture holds in its strong, completely boundedform, i.e., where p above is allowed to be any matrix-valued polynomial, forall tridiagonal 3×3 matrices with constant main diagonal: ⎡⎣⎢ac10b1ac20b2a⎤⎦⎥,a,bk,ck∈C, or equivalently, for all complex 3×3 matriceswith 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.

Original language	Undefined/Unknown
Publisher	arXiv.org
Publication status	Published - 2017
MoE publication type	D4 Published development or research report or study

Access to Document

https://arxiv.org/abs/1701.01365

Cite this

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title = "Crouzeix's conjecture holds for tridiagonal 3x3 matrices with elliptic numerical range centered at an eigenvalue",

abstract = "M. Crouzeix formulated the following conjecture in (Integral EquationsOperator Theory 48, 2004, 461--477): For every square matrix A and everypolynomial p, ∥p(A)∥≤2maxz∈W(A)|p(z)|, where W(A) is the numerical rangeof A. We show that the conjecture holds in its strong, completely boundedform, i.e., where p above is allowed to be any matrix-valued polynomial, forall tridiagonal 3×3 matrices with constant main diagonal: ⎡⎣⎢ac10b1ac20b2a⎤⎦⎥,a,bk,ck∈C, or equivalently, for all complex 3×3 matriceswith 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.",

author = "Christer Glader and Mikael Kurula and Mikael Lindstr{\"o}m",

year = "2017",

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TY - BOOK

T1 - Crouzeix's conjecture holds for tridiagonal 3x3 matrices with elliptic numerical range centered at an eigenvalue

AU - Glader, Christer

AU - Kurula, Mikael

AU - Lindström, Mikael

PY - 2017

Y1 - 2017

N2 - M. Crouzeix formulated the following conjecture in (Integral EquationsOperator Theory 48, 2004, 461--477): For every square matrix A and everypolynomial p, ∥p(A)∥≤2maxz∈W(A)|p(z)|, where W(A) is the numerical rangeof A. We show that the conjecture holds in its strong, completely boundedform, i.e., where p above is allowed to be any matrix-valued polynomial, forall tridiagonal 3×3 matrices with constant main diagonal: ⎡⎣⎢ac10b1ac20b2a⎤⎦⎥,a,bk,ck∈C, or equivalently, for all complex 3×3 matriceswith 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.

AB - M. Crouzeix formulated the following conjecture in (Integral EquationsOperator Theory 48, 2004, 461--477): For every square matrix A and everypolynomial p, ∥p(A)∥≤2maxz∈W(A)|p(z)|, where W(A) is the numerical rangeof A. We show that the conjecture holds in its strong, completely boundedform, i.e., where p above is allowed to be any matrix-valued polynomial, forall tridiagonal 3×3 matrices with constant main diagonal: ⎡⎣⎢ac10b1ac20b2a⎤⎦⎥,a,bk,ck∈C, or equivalently, for all complex 3×3 matriceswith 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.

M3 - Beställd rapport

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

PB - arXiv.org

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