Completely bounded analogue of Crouzeix’s conjecture

Resolve the completely bounded, or matrix-valued, analogue of Crouzeix’s conjecture concerning the universal spectral-set bound associated with the numerical range of a Hilbert-space operator.

Background

Crouzeix’s conjecture asserts that the closure of the numerical range of every bounded operator on a complex Hilbert space is a 2-spectral set, meaning that rational functions evaluated at the operator are bounded by twice their supremum on the numerical range. The paper notes that the scalar conjecture has recently received several independent proofs, while its completely bounded or matrix-valued extension remains unresolved.

The paper’s main theorem proves the Cloûatre–Ostermann–Ransford abstract conjecture, which is related to the scalar Crouzeix conjecture, but it does not address the completely bounded extension. Resolving that extension would establish the corresponding uniform estimate for matrix-valued or completely bounded functional calculi.