Verblunsky coefficients, CMV matrices and numerical invariants of homogeneous bidisc submodules

Abstract

Let $[p]$ be the principal submodule generated by a polynomial in $H^2(\mathbb D^2)$. For homogeneous $p$, the homogeneous slices of $[p]$ admit a weighted OPUC model in which the two wandering vectors are an orthonormal polynomial and its reversal. We show that the associated Verblunsky coefficients determine the singular values of the wandering-projection product and the restricted cross-commutator, as well as the non-zero spectrum of the core operator. Toeplitz-determinant and Mahler-measure identities yield exact Fredholm determinants and Schatten estimates, while $[(z-w)^N]$ rules out a uniform Hilbert--Schmidt bound. The same model gives explicit singular values of $[S_z^*,S_w]$ on the homogeneous quotient $H^2(\mathbb D^2)\ominus[p]$; for $p=(z-w)^N$, its squared Hilbert--Schmidt norm is asymptotic to $N$. For arbitrary polynomial generators, we construct a weighted bivariate model with a doubly Toeplitz, block-banded moment matrix and prove $C_p^2|_{\mathscr E_z}=Γ_p^*Γ_p$, relating the core spectrum to the cross-Gram operator between the two edge spaces. We also discuss cyclic-factor obstructions, represent higher numerical invariants by alternating CMV products, and give a quadratic counterexample to their proposed monotonicity.

Disclosure

“NVARIANTS 31 estimates depend on degree, separating the algebraic, Schatten, determinant and high-order aspects. AI disclosure. The research questions and main mathematical ideas in this article originated with the authors. ChatGPT 5.6 Sol (OpenAI) was then used to explore and verify these ideas through symbolic and numerical calculations. In particular, it found and verified the quadratic counterexample in Section 8. It also assisted with the exposition and language”

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Classification

Substantial mathematical content or result generation

Multiplier

10

Verified

2026-08-22T18:29:25.079390Z

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