The Crouzeix--Palencia bound for disk numerical ranges

The explicit nested exhaustion by concentric circles discharges every geometric and analytic premise of the nested scalar-companion assembly when the closed numerical range is a disk. On a disk the canonical closed scalar companion is the constant star (p(c)), so constant polynomials give exact approximation at every stage.

theorem crouzeix_palencia_of_closure_numericalRange_eq_closedBall {E : Type u} [NormedAddCommGroup E] [InnerProductSpace ℂ E] [CompleteSpace E] (A : E →L[ℂ] E) (c : ℂ) (R : ℝ) (hR : 0 ≤ R) (hW : closure (numericalRange A) = Metric.closedBall c R) : IsKPolynomialSpectralSet A (1 + √2) (closure (numericalRange A))

If the closed numerical range is a disk, it is a (1 + sqrt 2)-polynomial spectral set.