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## 2. $3\times3$ matrix and dilation theory

1. #### 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?
It is known that the implication holds if $A \in \mathbb{C}^{2\times 2}$ or $A \in \mathbb{C}^{3\times 3}$ of the form $[\lambda] \oplus A_0$ with $A_0 \in \mathbb{C}^{2\times 2}$ up to unitary similarity; see [MR0318920] [MR0253059] [MR1752164] [MR1870416]

1. T. Ando, Structure of operators with numerical radius one, Acta Sci. Math. (Seged) 34 (1971), 11-15.
2. A.W. Averson, Subalgebra of $C^*$-algebras, Acta Sci. Math. 123 (1969), 141-224.
3. M.D. Choi and C.K. Li, Numerical ranges and dilations, Linear and Multilinear Algebra 47 (2000), 35-48.
4. M.D. Choi and C.K. Li, Constrained Unitary Dilations and Numerical Ranges, J. Operator Theory, 46 (2001), 435–447
• #### Problem 2.2.

As $\Omega\to$ line segment, does $C_{\Omega}\to 1$?

Cite this as: AimPL: Crouzeix's conjecture, available at http://aimpl.org/crouzeix.