FPRD hunt packet: the complete Crouzeix similarity metric

Date: 8 September 2026
Mode: HUNT
Status: selected for ATTACK; no theorem claimed

Exact target

For an operator $T$ on a complex Hilbert space, let

$$W(T)=\{\langle Tx,x\rangle:\|x\|=1\}.$$

The scalar Crouzeix conjecture was proved in 2026: for every scalar polynomial $f$,

$$\|f(T)\|\le 2\sup_{z\in W(T)}|f(z)|.$$

The complete, or matrix-valued, form remains open. It asks whether, for every matrix-valued polynomial $F$ of every size,

$$\boxed{\ \|F(T)\|\le 2\sup_{z\in W(T)}\|F(z)\|.\ }$$

Equivalently, the evaluation homomorphism

$$\theta_T:A(\overline{W(T)})\longrightarrow B(\mathcal H), \qquad f\longmapsto f(T),$$

should have completely bounded norm at most two.

Why this is the right frontier

Lorist–Schwenninger, Jin, and Luo recently proved the scalar constant two by different routes. Badea–O’Loughlin–Virtanen then proved the associated abstract scalar theorem and explicitly recorded that the completely bounded analogue remains open. Crouzeix–Palencia’s older constant $1+\sqrt2$ is already completely bounded, so the open problem is exactly the sharp drop from $1+\sqrt2$ to $2$, not the existence of a dimension-free matrix bound.

This target is controlled by one global invariant rather than a family of matrix configurations. It also has a sharp positive-operator formulation which makes success and failure both structural.

The intrinsic object

Let $\Omega$ be a smooth bounded convex domain containing $\overline{W(T)}$, let $\varphi:\Omega\to\mathbb D$ be a Riemann map, and put $A=\varphi(T)$. Define the similarity radius

$$\gamma(A)=\inf\left\{ \sqrt{\kappa(P)}: P\succ0,\ A^*PA\preceq P \right\}, \qquad \kappa(P)=\frac{\lambda_{\max}(P)}{\lambda_{\min}(P)}.$$

Paulsen’s similarity theorem gives the exact equivalence

$$\|\theta_T\|_{\mathrm{cb}}\le2 \quad\Longleftrightarrow\quad \gamma(\varphi(T))\le2.$$

After rescaling $P$, the desired certificate is the single Stein inequality

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

One such metric controls every matrix amplification automatically. Shrinking smooth outer domains to $\overline{W(T)}$ gives the original problem. If the numerical range is a line segment, $T$ is an affine transform of a self-adjoint operator and the complete constant is one.

Independence of conformal normalization

Any two Riemann maps differ by a disk automorphism $m$. If $S^{-1}AS$ is a contraction, von Neumann’s inequality makes $m(S^{-1}AS)=S^{-1}m(A)S$ a contraction. Applying $m^{-1}$ gives the converse. Thus

$$\gamma(m(A))=\gamma(A),$$

so the similarity radius is intrinsic to $(T,\Omega)$ rather than a chosen coordinate.

Sharp positive control

For

$$N=\begin{pmatrix}0&2\\0&0\end{pmatrix}, \qquad W(N)=\overline{\mathbb D},$$

the metric $P=\operatorname{diag}(1,4)$ satisfies

$$N^*PN=\operatorname{diag}(0,4)\preceq P, \qquad \kappa(P)=4.$$

This is sharp. If $P=(p_{ij})\succ0$ and $N^*PN\preceq P$, then $p_{22}\ge4p_{11}$. Since $\lambda_{\max}(P)\ge p_{22}$ and $p_{11}\ge\lambda_{\min}(P)$, one has $\kappa(P)\ge4$. Hence $\gamma(N)=2$, exactly matching the sharp Crouzeix constant.

Known mechanism and the missing bridge

The scalar proofs produce a positive double-layer map. For a normalized scalar $f$, operator-valued Herglotz representation gives an isometry $V$ and a contraction (or unitary) $Q$ such that the defect moments

$$E_n=2V^*Q^{*n}V-A^{*n}$$

are uniformly bounded and commute with $A$. The Lorist–Schwenninger perturbation lemma converts that moment information into the scalar norm bound $\|A\|\le2$.

At matrix level, commutativity is lost after amplification. Directly amplifying the scalar perturbation argument is therefore not a route; the recent papers identify exactly this obstruction. Hartz–McCarthy’s completely bounded transfer works when the correction is scalar-valued, but the Crouzeix boundary correction is operator-valued.

The selected bridge is to use the full positive Toeplitz/Herglotz kernel, not pairwise commutativity, to construct one common metric $P$.

First ATTACK: common-metric transfer gate

The attack has three bounded steps.

1. Exact reduction. Write the complete spectral-set statement as the Stein certificate $I\preceq P\preceq4I$, $A^*PA\preceq P$, including the outer-domain limiting argument.
2. Positive-kernel extraction. Express the Herglotz moments as a positive block Toeplitz kernel and derive the precise Schur-complement condition which would produce such a $P$ without choosing a matrix level.
3. Universal transfer or countermodel. Either prove that positivity plus the scalar defect identities imply the metric, or build a finite operator-theoretic countermodel satisfying every retained axiom but with $\gamma(A)>2$. A countermodel stops this proof architecture without refuting complete Crouzeix.

The positive controls are the normalized $2\times2$ nilpotent above and the disk numerical-range case, where Berger dilation and Okubo–Ando similarity already yield the metric.

Explicit stopping rules

• Do not enumerate matrix sizes, Jordan forms, numerical-range shapes, or exceptional families.
• Do not fit a separate semidefinite certificate for individual matrices.
• Do not treat scalar norm agreement as a complete-bound proof.
• If the positive-kernel axioms admit a finite countermodel, record it and stop this route.
• If the only surviving path is a dimension-by-dimension matrix search, return to HUNT.

Priority and calibration

This packet selects a recognized open problem and provides an exact intrinsic reformulation plus a sharply falsifiable proof program. The similarity reformulation uses standard results of Paulsen and Okubo–Ando; it is not a new theorem. No advance toward the complete Crouzeix conjecture is claimed at HUNT stage, and no Publication is warranted.

Primary sources

1. C. Badea, R. O’Loughlin, and J. Virtanen, On the abstract approach to spectral constants: a proof of the Clouâtre–Ostermann–Ransford conjecture, arXiv:2609.03637v1 (3 September 2026). https://arxiv.org/html/2609.03637v1
2. E. Lorist and F. L. Schwenninger, A solution to Crouzeix’s conjecture, arXiv:2608.03841v2 (17 August 2026). https://arxiv.org/html/2608.03841v2
3. M. Hartz and J. E. McCarthy, From Clouâtre–Ostermann–Ransford to Okubo–Ando, arXiv:2606.02922v1 (2 June 2026). https://arxiv.org/html/2606.02922v1
4. M. Crouzeix and C. Palencia, The numerical range is a $(1+\sqrt2)$-spectral set, arXiv:1702.00668; SIAM J. Matrix Anal. Appl. 38 (2017), 649–655. https://arxiv.org/abs/1702.00668
5. K. R. Davidson, V. I. Paulsen, and H. J. Woerdeman, Complete spectral sets and numerical range, arXiv:1612.05683; Proc. Amer. Math. Soc. 146 (2018), 1189–1195. https://arxiv.org/abs/1612.05683

Next mode

ATTACK. Prove or refute the common-metric transfer lemma at the level of the positive Herglotz kernel. The first decisive output must be either a dimension-free construction of $P$ or an abstract finite countermodel to that construction.