Operator theory · completely bounded maps · selected target

The complete Crouzeix similarity metric

The open matrix-valued Crouzeix conjecture is equivalent to one positive metric with condition number at most four. An exact audit shows that extracting it from generic positive Herglotz data is already the open complete COR conjecture.

Recognized open problem

Can the sharp constant two hold at every matrix level?

For an operator $T$ and every matrix-valued polynomial $F$ , the complete Crouzeix conjecture asks for

$\boxed{\ \|F(T)\|\le2\sup_{z\in W(T)}\|F(z)\|.\ }$

Lorist and Schwenninger, Jin, and Luo recently proved the scalar inequality. Badea, O'Loughlin, and Virtanen prove the associated abstract scalar theorem and explicitly identify the completely bounded analogue as open. The older $1+\sqrt2$ estimate of Crouzeix and Palencia is already complete.

Intrinsic reformulation

One positive metric controls all amplifications

Let $\Omega\supset\overline{W(T)}$ be smooth and convex, choose a Riemann map $\varphi:\Omega\to\mathbb D$ , and put $A=\varphi(T)$ . Define

$\gamma(A)=\inf_{P\succ0,\ A^*PA\preceq P}\sqrt{\frac{\lambda_{\max}(P)}{\lambda_{\min}(P)}}.$

Paulsen's similarity theorem gives the exact complete-bound target

$\boxed{\ I\preceq P\preceq4I,\qquad A^*PA\preceq P.\ }$

Disk automorphisms preserve this infimum in both directions by von Neumann's inequality. Thus the object is independent of the Riemann-map normalization. Shrinking outer domains recovers the numerical range itself. This packages every matrix size into one dimension-independent Stein certificate.

Sharp positive control

The extremal nilpotent forces condition number four

For $N=\begin{pmatrix}0&2\\0&0\end{pmatrix},\qquad W(N)=\overline{\mathbb D},$ the metric $P=\operatorname{diag}(1,4)$ obeys $N^*PN\preceq P$ . Conversely, the lower-right Stein entry forces $p_{22}\ge4p_{11}$ , hence every feasible metric has condition number at least four. Therefore $\gamma(N)=2$ .

Exact stopping boundary

The proposed transfer is the complete COR conjecture

For a unital completely bounded homomorphism $\theta$ and an antilinear complete contraction $\alpha$ , put

$\Lambda(f)=\theta(f)+\theta(\alpha(f))^*.$

For the convex double-layer map, $\Lambda/2$ is already unital completely positive. Its full matricial Herglotz and block Toeplitz positivity therefore repackages the hypothesis $\|\Lambda\|_{\rm cb}\le2$ . Deducing the metric above is, by Paulsen similarity, precisely the open complete COR implication

$\|\Lambda\|_{\rm cb}\le2\quad\Longrightarrow\quad\|\theta\|_{\rm cb}\le2.$

Thus Toeplitz minors and Schur complements supply no smaller lemma. A finite abstract countermodel to the retained axioms would refute complete COR, rather than merely stopping this proof. The route now stops unless a geometry-specific resolvent identity adds information beyond the abstract hypotheses.

Sources and scope

The scalar breakthrough isolates the complete obstruction

Badea–O'Loughlin–Virtanen (2026): abstract sharp scalar theorem, positivity, and Herglotz dilation.
Lorist–Schwenninger (2026): scalar perturbation lemma and sharp numerical-range theorem.
Hartz–McCarthy (2026): completely bounded scalar-correction transfer and the Okubo–Ando connection.
Jury–Tsikalas (2023): double-layer positivity and the distinction between contractive and completely contractive spectral-set classes.
Davidson–Paulsen–Woerdeman (2018): complete spectral sets, complete numerical radius, and similarity.

The similarity formulation remains useful, but the generic positive-kernel attack is circular at the exact open implication. Complete Crouzeix remains open; this stopping audit is not a Publication.