arXiv:2607.10507 2026 Dynamics 1 ideas extracted · analyzed Aug 30, 2026
The paper supplies a norm-gap principle for detecting analyticity from bounded operator polynomials: a strict inequality between the operator norm of a polynomial and its unit-circle boundary norm forces strong functional regularity. This suggests a practical regularizer for recurrent, state-space, and deep equilibrium networks that discourages transition operators with pathological powers or near-unit-circle pseudospectral behavior, rather than merely penalizing eigenvalues. The most promising implementation is to evaluate a small bank of polynomial probes on the learned transition matrix, estimate their spectral norms with power iteration, and impose a soft Beurling–Kato margin while preserving task loss. The geometric contour lemma is less directly useful than the polynomial norm-gap mechanism.
Add a functional-calculus regularizer to the transition operator of an RNN, linear state-space model, or deep-equilibrium layer. The regularizer uses polynomial probes to detect non-normal transient amplification that ordinary eigenvalue-radius penalties can miss.