Crouzeix's conjecture holds for a subclass of tridiagonal 3x3 matrices

Christer Glader; Mikael Kurula; Mikael Lindström

Crouzeix's conjecture holds for a subclass of tridiagonal 3x3 matrices

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 thenumerical range of A. We show that the conjecture holds for tridiagonal3×3 matrices ⎡⎣⎢ac0bac0ba⎤⎦⎥,a,b,c∈C, with constant diagonals, even if anynumber of off-diagonal elements should be set to zero. We also extend the mainresult 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/1701.01365v1

Cite this

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title = "Crouzeix's conjecture holds for a subclass of tridiagonal 3x3 matrices",

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 thenumerical range of A. We show that the conjecture holds for tridiagonal3×3 matrices ⎡⎣⎢ac0bac0ba⎤⎦⎥,a,b,c∈C, with constant diagonals, even if anynumber of off-diagonal elements should be set to zero. We also extend the mainresult of D. Choi in (Linear Algebra Appl. 438, 3247--3257) slightly.",

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T1 - Crouzeix's conjecture holds for a subclass of tridiagonal 3x3 matrices

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 thenumerical range of A. We show that the conjecture holds for tridiagonal3×3 matrices ⎡⎣⎢ac0bac0ba⎤⎦⎥,a,b,c∈C, with constant diagonals, even if anynumber of off-diagonal elements should be set to zero. We also extend the mainresult 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 thenumerical range of A. We show that the conjecture holds for tridiagonal3×3 matrices ⎡⎣⎢ac0bac0ba⎤⎦⎥,a,b,c∈C, with constant diagonals, even if anynumber of off-diagonal elements should be set to zero. We also extend the mainresult of D. Choi in (Linear Algebra Appl. 438, 3247--3257) slightly.

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PB - arXiv.org

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