Square Functions, 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. We also obtain sharp column and row square-function inequalities, 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.