The Complete Crouzeix Conjecture in Dimension Three and the Clouâtre-Ostermann-Ransford conjecture
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.
Related
- On the abstract approach to spectral constants: a proof of the Clouâtre--Ostermann--Ransford conjecture
- A short proof of the complete Crouzeix inequality in dimension three
- Arithmetic purity of strong approximation for complete intersections of large dimension
- A Complete Solution to the Conjecture on the Inverse Collatz–Sinogowitz Irregularity Problem
- Bounded VC-dimension implies the Erdős--Rado sunflower conjecture
BibTeX
@misc{ahag2026complete,
title = {{The Complete Crouzeix Conjecture in Dimension Three and the Clouâtre-Ostermann-Ransford conjecture}},
author = {Per Åhag and Rafał Czyż and Jani Virtanen},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2608.27346},
eprint = {2608.27346},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.27346},
}
RIS
TY - UNPB
TI - The Complete Crouzeix Conjecture in Dimension Three and the Clouâtre-Ostermann-Ransford conjecture
AU - Per Åhag
AU - Rafał Czyż
AU - Jani Virtanen
PY - 2026
DA - 2026/09/21
PB - arXiv
DO - 10.48550/arxiv.2608.27346
UR - https://arxiv.org/abs/2608.27346
AB - 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.
ER -