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On the abstract approach to spectral constants: a proof of the Clouâtre--Ostermann--Ransford conjecture
Abstract
Clouâtre, Ostermann, and Ransford formulated an abstract version of Crouzeix's conjecture involving a bounded unital homomorphism from a uniform algebra into matrices and a unital antilinear map. They conjectured that contractivity of the associated symmetrised map forces the homomorphism to have norm at most two. We prove this conjecture, in fact without requiring the antilinear map to be contractive and for homomorphisms into the bounded operators on a Hilbert space. The proof combines positivity of real parts, an operator-valued Herglotz theorem, and the perturbation lemma of Lorist and Schwenninger used in the recent proof of Crouzeix's conjecture.
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BibTeX
@misc{badea2026abstract,
title = {{On the abstract approach to spectral constants: a proof of the Clouâtre--Ostermann--Ransford conjecture}},
author = {Catalin Badea and Ryan O'Loughlin and Jani Virtanen},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.03637},
eprint = {2609.03637},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.03637},
}
RIS
TY - UNPB
TI - On the abstract approach to spectral constants: a proof of the Clouâtre--Ostermann--Ransford conjecture
AU - Catalin Badea
AU - Ryan O'Loughlin
AU - Jani Virtanen
PY - 2026
DA - 2026/09/03
PB - arXiv
DO - 10.48550/arxiv.2609.03637
UR - https://arxiv.org/abs/2609.03637
AB - Clouâtre, Ostermann, and Ransford formulated an abstract version of Crouzeix's conjecture involving a bounded unital homomorphism from a uniform algebra into matrices and a unital antilinear map. They conjectured that contractivity of the associated symmetrised map forces the homomorphism to have norm at most two. We prove this conjecture, in fact without requiring the antilinear map to be contractive and for homomorphisms into the bounded operators on a Hilbert space. The proof combines positivity of real parts, an operator-valued Herglotz theorem, and the perturbation lemma of Lorist and Schwenninger used in the recent proof of Crouzeix's conjecture.
ER -