2. $3\times3$ matrix and dilation theory

Problem 2.1.
[C.K. Li and Y.T. Poon] For which $A \in \mathbb{C}^{3\times 3}$ holds that $W(T) \subseteq W(A)$ implies that $T = X^*(I\otimes A)X$ for some $X^*X = I$, i.e., $I\otimes A$ is a dilation of $T$? More generally, for which matrix or operator $A$ does the implication hold?
 T. Ando, Structure of operators with numerical radius one, Acta Sci. Math. (Seged) 34 (1971), 1115.
 A.W. Averson, Subalgebra of $C^*$algebras, Acta Sci. Math. 123 (1969), 141224.
 M.D. Choi and C.K. Li, Numerical ranges and dilations, Linear and Multilinear Algebra 47 (2000), 3548.
 M.D. Choi and C.K. Li, Constrained Unitary Dilations and Numerical Ranges, J. Operator Theory, 46 (2001), 435–447
Cite this as: AimPL: Crouzeix's conjecture, available at http://aimpl.org/crouzeix.