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

Abstract: Let $[p]$ be the principal submodule generated by a polynomial in $H^2(\mathbb D<sup>2)$. 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 $[(z-w)<sup>N]$ rules out a uniform Hilbert--Schmidt bound. The same model gives explicit singular values of $[S_z<sup>*,S_w]$ on the homogeneous quotient $H^2(\mathbb D<sup>2)\ominus[p]$; for $p=(z-w)<sup>N$, 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 $C_p^2|_{\mathscr E_z}=Γ_p<sup>*Γ_p$, 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.

• The paper establishes a complete OPUC-based spectral dictionary in which Verblunsky coefficients determine the singular values, spectra, and Schatten-class behavior of projection products, cross-commutators, and core operators for homogeneous principal submodules.
• It derives exact Toeplitz-determinant and Mahler-measure formulas, proves the sharp degree bound \(\|X_{[p]}\|_{HS}^2<2d\log 2\), and shows that the Hilbert–Schmidt norms for \((z-w)^N\) grow asymptotically like \(2N\).
• It demonstrates that higher invariants need not decrease with the order—an explicit quadratic generator violates monotonicity—while quotient and nonhomogeneous cases require additional spectral-factor or bivariate orthogonal-polynomial data.

This paper develops a complete operator-theoretic dictionary between the classical theory of orthogonal polynomials on the unit circle (OPUC) and the numerical invariants of principal submodules of the Hardy module $H^2(\mathbb D^2)$ generated by homogeneous polynomials. Its central observation is that for a homogeneous generator $p$, the slice-wise wandering vectors of the submodule $[p]$ are precisely an orthonormal polynomial and its reversal for an explicit weighted measure on $\mathbb T$, so that Verblunsky coefficients encode all standard defect operators: the projection product $P_zP_w$, the cross-commutator $X_{[p]}=[R_z^*,R_w]$, and the core operator $C_{[p]}$. The paper then extends this picture to quotient modules, higher invariants via CMV matrices, quasi-homogeneous generators, and — in a necessarily weaker form — arbitrary polynomial generators.

The OPUC model for homogeneous submodules

For $p(z,w)=\sum c_j z^jw^{d-j}$ homogeneous and $q(\zeta)=p(\zeta,1)$, multiplication by $p$ identifies the $p$0th homogeneous slice $p$1 with polynomials of degree at most $p$2 in $p$3, where $p$4. Under this unitary map, the wandering spaces $p$5 and $p$6 have orthonormal bases given by the reversed orthonormal polynomial $p$7 and $p$8 respectively, homogenized slice by slice. Two elementary overlaps from the Szegő recursions drive everything:

$p$9

The main structural result is then a complete spectral dictionary: the non-zero singular values of $[p]$0 are $[p]$1; those of $[p]$2 are $[p]$3; and the non-zero spectrum of $[p]$4 is $[p]$5. Consequently $[p]$6 if and only if $[p]$7, with $[p]$8. Since every non-zero polynomial weight satisfies the Szegő condition, every homogeneous principal polynomial submodule is Hilbert–Schmidt, with

$[p]$9

Baxter's theorem further identifies trace class membership: $\mathbb T$0 is trace class exactly when $\mathbb T$1 has no zero on $\mathbb T$2, in which case the coefficients decay exponentially.

Determinant identities and degree bounds

Toeplitz determinant ratios give each singular value explicitly via Desnanot–Jacobi: $\mathbb T$3, where $\mathbb T$4 are Toeplitz determinants of $\mathbb T$5. Globally, the Szegő–Verblunsky product theorem yields exact Fredholm determinants,

$\mathbb T$6

with $\mathbb T$7 the Mahler measure, together with a logarithmic trace identity summing $\mathbb T$8. A sharp coefficient-versus-Mahler-measure inequality, $\mathbb T$9 with equality exactly for powers of a toral linear factor, gives the degree bound $P_zP_w$0. This bound is attained asymptotically on the family $P_zP_w$1: the pure Fisher–Hartwig weight has Verblunsky coefficients $P_zP_w$2, giving

$P_zP_w$3

Hence core Hilbert–Schmidt norms are unbounded even among homogeneous principal polynomial submodules, answering Problem 10 of Yang's survey negatively. The Fredholm determinant here equals $P_zP_w$4, showing the Mahler-measure lemma is sharp while the passage from determinant to Hilbert–Schmidt norm is strict.

Quotient modules and compressed cross-commutators

On the quotient $P_zP_w$5, the compressed shifts need not be isometries, and the mechanism changes: from the doubly commuting ambient shifts one obtains $P_zP_w$6 versus $P_zP_w$7, an "$P_zP_w$8 versus $P_zP_w$9" phenomenon. Each degree block of the compressed cross-commutator has rank at most one, with singular values

$X_{[p]}=[R_z^*,R_w]$0

where $X_{[p]}=[R_z^*,R_w]$1 are normalized extreme coefficients of $X_{[p]}=[R_z^*,R_w]$2 and $X_{[p]}=[R_z^*,R_w]$3 are leading coefficients. The extra factors measure how much of each wandering vector survives a backward shift; notably, the Verblunsky sequence alone does not determine the quotient cross-commutator — the choice of spectral factor enters through $X_{[p]}=[R_z^*,R_w]$4. For monomial generators this yields a sharp dichotomy: rank-zero when the monomial is pure, rank-one norm-one otherwise. For $X_{[p]}=[R_z^*,R_w]$5 the formula recovers the Bergman self-commutator diagonal and the exact value $X_{[p]}=[R_z^*,R_w]$6. For $X_{[p]}=[R_z^*,R_w]$7, Stirling-based Riemann-sum estimates give $X_{[p]}=[R_z^*,R_w]$8, so no uniform Hilbert–Schmidt bound exists for quotient cross-commutators either; the paper poses determining the sharp constant $X_{[p]}=[R_z^*,R_w]$9 as an open problem.

Higher invariants, CMV products, and failure of monotonicity

The higher numerical invariants $C_{[p]}$0 reduce to $C_{[p]}$1-energies of the sequences $C_{[p]}$2, which admit a CMV representation: $C_{[p]}$3 and $C_{[p]}$4 are parity-diagonal entries of the unitary words $C_{[p]}$5 and $C_{[p]}$6. Unitarity alone cannot force monotonicity in $C_{[p]}$7, since different words contribute different diagonals and off-diagonal mass can return.

This obstruction is realized concretely. For the quadratic $C_{[p]}$8, the Schur algorithm produces a six-step periodic rational parametrization of the Verblunsky sequence, and exact alternating-CMV computation — certified by explicit integer polynomials in an appendix, with no numerical approximation — gives

$C_{[p]}$9

Thus the proposed monotone decrease of $p(z,w)=\sum c_j z^jw^{d-j}$0 fails already for a quadratic homogeneous generator; earlier monotonicity computations for $p(z,w)=\sum c_j z^jw^{d-j}$1 and $p(z,w)=\sum c_j z^jw^{d-j}$2 reflect special Verblunsky data rather than a general principle. The paper leaves open the full phase diagram of $p(z,w)=\sum c_j z^jw^{d-j}$3 for the family $p(z,w)=\sum c_j z^jw^{d-j}$4.

Quasi-homogeneous generators and the general case

For $p(z,w)=\sum c_j z^jw^{d-j}$5-quasi-homogeneous $p(z,w)=\sum c_j z^jw^{d-j}$6 with $p(z,w)=\sum c_j z^jw^{d-j}$7, the wandering spaces split into residue classes under the character $p(z,w)=\sum c_j z^jw^{d-j}$8; all mixed pairings vanish except in the common zero class. Hence the non-zero singular values coincide with those of the ordinary homogeneous reduction $p(z,w)=\sum c_j z^jw^{d-j}$9 — the weights and monomial factor add only orthogonal zero blocks, so quasi-homogeneity yields no new spectral data.

Beyond the graded setting, the scalar OPUC model collapses. Multiplication by $q(\zeta)=p(\zeta,1)$0 still gives a unitary equivalence between $q(\zeta)=p(\zeta,1)$1 and the polynomial closure in $q(\zeta)=p(\zeta,1)$2 on $q(\zeta)=p(\zeta,1)$3, but distinct total-degree layers are mutually orthogonal if and only if $q(\zeta)=p(\zeta,1)$4 is homogeneous; otherwise the moment matrix is only block banded (bandwidth $q(\zeta)=p(\zeta,1)$5), though always doubly Toeplitz with finite Fourier bandwidth. In this bivariate model the paper proves the exact identity

$q(\zeta)=p(\zeta,1)$6

where $q(\zeta)=p(\zeta,1)$7 is the cross-Gram operator between the two edge spaces — the canonical correlations between them. Thus the core spectrum of any polynomial principal submodule is governed by bivariate orthogonal polynomials, reducing to the scalar Verblunsky formula precisely in the homogeneous case.

Two obstructions prevent verbatim extension of the homogeneous formulas. First, cyclic factors are invisible: $q(\zeta)=p(\zeta,1)$8 for cyclic $q(\zeta)=p(\zeta,1)$9, yet for $p$0 the Mahler ratio $p$1 disagrees with the actual Fredholm determinant $p$2. A cyclically invariant determinant formula remains open. Second, there is as yet no canonical, ordering-independent block-CMV realization of the finite-band moment matrix intrinsic to the submodule rather than the chosen generator.

Limitations and open problems

The paper is candid about scope. The complete Verblunsky dictionary holds only for homogeneous generators; the general model is exact but two-variable, with no single scalar measure or coefficient sequence. Compactness of the core is guaranteed for polynomial submodules but assumed conditionally in the general weighted-model statement, since non-polynomial generators may fail it. The counterexample to monotonicity is a single parameter value verified by computer-algebra certificates rather than a uniform analytic argument. The closing section formulates six precise problems: a canonical block model with cyclic-factor-invariant recurrence parameters; sharp asymptotics of the quotient degree constant $p$3 (is $p$4, and are toral linear-factor powers extremal?); Schatten thresholds $p$5 in terms of toral zero geometry; a cyclically invariant determinant $p$6 as a limit of finite sections; the quadratic phase diagram; and large-$p$7 behavior of $p$8 for multiple toral factors and of the intrinsic operator $p$9 in the general case.

Conclusion

The paper establishes that, for homogeneous principal submodules of the bidisc Hardy module, the entire apparatus of numerical invariants — projection products, cross-commutators, core spectra, Schatten membership, Fredholm determinants, and higher invariants — is governed by a single Verblunsky sequence, with exact formulas validated against Fisher–Hartwig asymptotics and the Bergman shift. It simultaneously demonstrates the rigidity of this reduction (orthogonality of degree layers characterizes homogeneity), the genuine failure of proposed monotonicity for higher invariants, and the two structural obstructions — degree coupling and cyclic factors — that any extension beyond the homogeneous class must confront.