MathDB

Complete Crouzeix conjecture

For every n,m≥1n,m\ge 1, every matrix A∈Mn(C)A\in M_n(\mathbb{C}), and every matrix-valued polynomial P(z)=∑k=0dPkzkP(z)=\sum_{k=0}^d P_k z^k with Pk∈Mm(C)P_k\in M_m(\mathbb{C}), one has ∥∑k=0dAk⊗Pk∥≤2sup⁡z∈W(A)∥P(z)∥\left\|\sum_{k=0}^d A^k\otimes P_k\right\|\le 2\sup_{z\in W(A)}\|P(z)\|, where W(A)={⟨Ax,x⟩:x∈Cn, ∥x∥=1}W(A)=\{\langle Ax,x\rangle:x\in\mathbb{C}^n,\ \|x\|=1\} is the numerical range of AA and the norms are operator norms.

References

Primary source

arXiv

Additional references

matrix-analysisoperator-theory
Posted

Progress summary

Refreshed
Claimed progress

A recent paper claims the complete conjecture in dimensions one through three, while the general problem remains open.

The complete Crouzeix conjecture asks for the conjectured bound at all matrix levels. The latest report claims the first nontrivial dimensions, through dimension three, together with rigidity and representing-measure results; no source independently verifies this claim.

Known results

The conjecture is established for normal matrices, 2×22\times2 matrices, selected tridiagonal matrices, certain contractions, and other specialized classes. A weighted-shift result proves the complete bound for matrices of the form M=Pddiag⁡(α1,…,αd)M=P_d\operatorname{diag}(\alpha_1,\ldots,\alpha_d), not in general.

August 2026 developments

A news item dated August 27, 2026 reports a paper proving the complete bound in dimensions one through three; this remains unverified. Separately, a July 2026 preprint claims the ordinary, not completely bounded, conjecture in full generality and explicitly says its method does not establish the complete version.

Current status (as of August 2026): The complete conjecture is claimed for dimensions one through three, but that result is unverified and the general case beyond dimension three remains open.

Sources

Solutions 0

No solutions have been posted yet.