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

@article{Glader2018CrouzeixsCH, title={Crouzeix's Conjecture Holds for Tridiagonal 3 x 3 Matrices with Elliptic Numerical Range Centered at an Eigenvalue}, author={Christer Glader and Mikael Kurula and M. Lindstr{\"o}m}, journal={SIAM J. Matrix Anal. Appl.}, year={2018}, volume={39}, pages={346-364} }

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...

#### 6 Citations

Crouzeix's Conjecture and related problems

- Mathematics, Computer Science
- ArXiv
- 2020

The conjecture that, for any polynomial f and any square matrix A, the operator norm of f(A) satisfies the estimate of Crouzeix’s conjecture is asserted and bounds for for specific classes of operators A are derived. Expand

A bivariate extension of the Crouzeix-Palencia result with an application to Fréchet derivatives of matrix functions

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- ArXiv
- 2020

The purpose of this work is to point out that the spectral norm of the Fr\'echet derivative of f(A) is bounded by $K^2$ times the maximum of $f^\prime$ on $W (A)$. Expand

Numerical range and compressions of the
shift

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- 2018

The numerical range of a bounded, linear operator on a Hilbert space is a set in $\mathbb{C}$ that encodes important information about the operator. In this survey paper, we first consider numerical… Expand

An abstract approach to the Crouzeix conjecture

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Let $A$ be a uniform algebra and let $\theta:A\to M_n(\mathbb{C})$ be a continuous homomorphism. Suppose that there exists an antilinear contraction $\alpha:A\to A$ such that \[… Expand

Some Extensions of the Crouzeix-Palencia Result

- Mathematics, Computer Science
- SIAM J. Matrix Anal. Appl.
- 2018

This paper shows how the arguments in Crouzeix and Palencia's paper can be extended to show that other regions in the complex plane that do not necessarily contain $W(A)$ are $K-spectral sets for a value of $K$ that may be close to $1 + \sqrt{2}$. Expand

On the uniqueness of functions that maximize the Crouzeix ratio

- Mathematics
- 2020

Let $A$ be an $n$ by $n$ matrix with numerical range $W(A) := \{ q^{*}Aq : q \in \mathbb{C}^n , ~\| q \|_2 = 1 \}$. We are interested in functions $\hat{f}$ that maximize $\| f(A) \|_2$ (the matrix… Expand

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