Journal of Operator Theory

Volume 90, Issue 1, Summer 2023 pp. 209-221.

An abstract approach to the Crouzeix conjecture

Authors: Raphael Clouatre, Maleva Ostermann

$2$ , and Thomas Ransford

$3$
Author institution:

$1$ Department of Mathematics, University of Manitoba, Winnipeg

$Manitoba$ , R3T 2N2, Canada

$2$ Departement de mathematiques et de statistique, Universite Laval, Quebec City

$Quebec$ , G1V 0A6, Canada

$3$ Departement de mathematiques et de statistique, Universite Laval, Quebec City

$Quebec$ , G1V 0A6, Canada

Summary: Let

$A$ be a uniform algebra,

$\theta:A\to M_n(\mathbb{C})$ be a continuous homomorphism and

$\alpha:A\to A$ be an antilinear contraction such that

$\|\theta(f)+\theta(\alpha(f))^*\|\leqslant 2\|f\| \quad(f\in A).$ We show that

$\|\theta\|\leqslant 1+\sqrt{2}$ , and that

$1+\sqrt2$ is sharp. We conjecture that, if further

$\alpha(1)=1$ , then we may conclude that

$\|\theta\|\leqslant 2$ . This would yield a positive solution to the Crouzeix conjecture on numerical ranges. In support of our conjecture, we prove that it is true in two special cases. We also discuss a completely bounded version of our conjecture that brings into play ideas from dilation theory.

DOI: http://dx.doi.org/10.7900/jot.2021nov15.2364
Keywords: Crouzeix conjecture, uniform algebra, homomorphism, completely bounded map.

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