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 $[S_z^{\ast },S_w]$ 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 $C_p^2{|}_{{\mathcal{E}}_z}={\Gamma }_p^{\ast }{\Gamma }_p$, 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.

35 pages