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arXiv · 2608.27346

The Complete Crouzeix Conjecture in Dimension Three and the Clouâtre-Ostermann-Ransford conjecture

Per Åhag·Rafał Czyż·Jani Virtanen
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Abstract

We settle the complete Crouzeix conjecture for matrices of order at most three and prove the Q-algebra case of the completely bounded Clouâtre--Ostermann--Ransford (COR) conjecture for homomorphisms into matrices of order at most three, via sharp abstract column and row estimates. Our approach also establishes the scalar COR conjecture in a stronger form, for homomorphisms with commutative range on Banach algebras with unity satisfying von Neumann's inequality. Under contractivity of the symmetrized map, this result holds on arbitrary Hilbert spaces without initial boundedness assumptions on the homomorphism or the antilinear map. For arbitrary operator algebras, we disprove the complete COR conjecture by an exact three-dimensional example with target matrix order two. We also prove the sharp complete bound for every matrix subalgebra containing the diagonal, in arbitrary matrix order and on arbitrary target Hilbert spaces. We also obtain column and row square-function inequalities with sharp norm bounds, strict scalar bounds for operators similar to normal operators, sharp complete spectral constants for scaled $q$-numerical ranges in dimensions two and three, and rigidity, stability, and representing-measure results.

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Per Åhag, Rafał Czyż, Jani Virtanen. 2026-09-21. The Complete Crouzeix Conjecture in Dimension Three and the Clouâtre-Ostermann-Ransford conjecture. https://arxiv.org/abs/2608.27346

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