Canonical block-CMV model for general polynomial generators

Construct canonical recurrence parameters from the doubly Toeplitz moment matrix associated with a general polynomial generator that determine the two defect projections, the intrinsic higher-invariant operators, and the core operator; determine whether a block-CMV or finite-band unitary realization exists that reduces to the scalar CMV matrix in the homogeneous case and identify an ordering that makes the parameters intrinsic to the principal submodule rather than its chosen generator.

Background

For homogeneous generators, a scalar orthogonal-polynomial model and a single Verblunsky sequence describe the wandering spaces, cross-commutator, core, and higher numerical invariants. For general polynomial generators, total-degree layers are coupled within a finite bandwidth, and the associated moment matrix is doubly Toeplitz rather than scalar Toeplitz.

The paper identifies the cross-Gram operator between the two edge spaces as the object governing the nontrivial core spectrum, but does not provide canonical recurrence data or a block-CMV realization for it. Any such construction must also be invariant under multiplication of the generator by a cyclic polynomial, since cyclic factors do not change the principal submodule.

References

Starting from the doubly Toeplitz matrix $G_p$ in eq:general-moment-formula, construct canonical recurrence parameters that determine $Q_z$, $Q_w$, $B_k(p)$ and the core operator. Is there a block-CMV or finite-band unitary realization which reduces to the scalar CMV matrix of Section~\ref{sec:cmv} when $p$ is homogeneous? Determine which lexicographic, reverse-lexicographic or total-degree ordering makes the resulting parameters intrinsic to the submodule rather than to the chosen generator.

eq:general-moment-formula:

Gp(α,β)=1∥p∥22∑γ,δ∈Aα+γ=β+δcγcδ‾.G_p(\alpha,\beta) =\frac1{\|p\|_2^2} \sum_{\substack{\gamma,\delta\in A\\ \alpha+\gamma=\beta+\delta}} c_\gamma\overline{c_\delta}.

— Verblunsky coefficients, CMV matrices and numerical invariants of homogeneous bidisc submodules  (2608.18456 - Lu et al., 19 Aug 2026) in Problem 1, Section 10, "Problems and further directions"

Assume that $X_{[p]}*X_{[p]}$ is trace class. Find an explicit quantity $\mathcal D([p])$, computable from $p$ but invariant under $p\mapsto up$ for every cyclic polynomial $u$, such that

\det(I-X_{[p]}*X_{[p]})=\mathcal D([p]).

Can $\mathcal D([p])$ be represented as a limit of determinants of finite sections of $G_p$?

— Verblunsky coefficients, CMV matrices and numerical invariants of homogeneous bidisc submodules  (2608.18456 - Lu et al., 19 Aug 2026) in Problem 4, Section 10, "Problems and further directions"