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3×33\times3 matrix and dilation theory · Question 2.1

Crouzeix's conjecture · AIMPL workshop record

MSC 47 · Operator theory

Editorial mapping of AIMPL list topic: Crouzeix's conjecture. http://aimpl.org/crouzeix/ Not an author-supplied MSC code.

Workshop record · status unassessedAs of 2026-09-10 source snapshot; not a current-status certificationSource & evidence ↓
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Problem statement

For which A∈C3×3A \in \mathbb{C}^{3\times 3} holds that W(T)⊆W(A)W(T) \subseteq W(A) implies that T=X∗(I⊗A)XT = X^*(I\otimes A)X for some X∗X=IX^*X = I, i.e., I⊗AI\otimes A is a dilation of TT? More generally, for which matrix or operator AA does the implication hold?

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Dated source record, inspected 2026-09-10. Current status has not been independently reassessed. Source-reported results have not been independently verified.

Status note in the source

It is known that the implication holds if A∈C2×2A \in \mathbb{C}^{2\times 2} or A∈C3×3A \in \mathbb{C}^{3\times 3} of the form
[λ]⊕A0[\lambda] \oplus A_0 with A0∈C2×2A_0 \in \mathbb{C}^{2\times 2} up to unitary similarity; see Source citation: MR0318920 Source citation: MR0253059 Source citation: MR1752164 Source citation: MR1870416

  • T. Ando, Structure of operators with numerical radius one, Acta Sci. Math. (Seged) 34 (1971), 11-15.

  • A.W. Averson, Subalgebra of C∗C^*-algebras, Acta Sci. Math. 123 (1969), 141-224.

  • M.D. Choi and C.K. Li, Numerical ranges and dilations, Linear and Multilinear Algebra 47 (2000), 35-48.

  • M.D. Choi and C.K. Li, Constrained Unitary Dilations and Numerical Ranges, J. Operator Theory, 46 (2001), 435--447

References
  1. [1]Crouzeix's conjecture · Original question link
  2. [2]AIMPL source bibliography link
  3. [3]AIMPL text license · CC BY-SA 3.0 linkAIM Problem Lists (AIMPL), original contributors; CC BY-SA 3.0. Adaptation: structural TeX converted to Markdown, documented source macros expanded, and source bibliography links added. Mathematical wording retained. These adapted source texts remain under CC BY-SA 3.0.

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