谱常数的抽象方法：Clouâtre–Ostermann–Ransford猜想的证明
On the abstract approach to spectral constants: a proof of the Clouâtre--Ostermann--Ransford conjecture
AI总结：
本文证明了Clouâtre–Ostermann–Ransford提出的谱常数抽象版本猜想，其无需反线性映射收缩，结合多种定理与引理完成证明，拓展了Crouzeix猜想的相关研究。
AI中文摘要：
Clouâtre、Ostermann和Ransford提出了Crouzeix猜想的抽象版本，涉及从一致代数到矩阵的有界单位同态及单位反线性映射，他们猜想相关对称映射的收缩性迫使同态的范数至多为2。本文证明了该猜想，且无需反线性映射具有收缩性，适用于从一致代数到希尔伯特空间上有界算子的同态。证明结合了实部正性、算子值Herglotz定理，以及Lorist和Schwenninger在近期Crouzeix猜想证明中使用的扰动引理。
英文摘要：
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.
