Testing von Neumann inequalities with nilpotent matrices

TL;DR

The paper provides an elementary cone-separation proof that the Agler norm $||p||_A$ of a polynomial is determined by testing on commuting contractive nilpotent $d$-tuples, reducing the problem to finite-dimensional data. It couples this reduction with the $d$-variable Carathéodory-Fejér interpolation framework of Eschmeier-Patton-Putinar to derive four equivalent formulations: a direct operator-inequality condition, a positive semidefinite matrix factorization, a modulus identity modulo the truncation $(z)^{N+1},(ar z)^{N+1}$, and a rational inner Agler decomposition. A key novelty is proving (1) ⇒ (2) via a closed cone $\\mathcal{C}$ and a separating hyperplane argument, enabling a construction of simple $N$-nilpotent contractions that witness failures of the inequality. The work further extends the framework to finite interpolation on a set $S \\\subset \\mathbb{D}^d$ with simultaneous diagonalizability, showing equivalence between interpolation feasibility and PSD-parameterizations, and connects to known counterexamples (e.g., Lotto–Steger) and prior interpolation theories (BLTT, EPP). Overall, it clarifies how rigorous von Neumann-type inequalities can be verified by finite-dimensional tests, with significant implications for both theory and explicit computations in multivariable operator interpolation.

Abstract

We give an elementary proof of the folklore result that the Agler norm of a function is determined by its norm on commuting tuples of nilpotent matrices. The proof is variation on a standard cone separation argument. The topic is closely related to the Eschmeier-Patton-Putinar formulation of Carathéodory-Fejér interpolation.

Theorems & Definitions (14)

• Theorem 1.1
• Definition 1.2
• Theorem 1.3
• Remark 1.4
• Theorem 1.5
• Remark 1.6
• proof : Proof of Theorem \ref{['genthm']} from Theorem \ref{['nilthm']}
• proof : Proof of Theorem \ref{['nilthm']}
• Lemma 2.1
• proof