Strict Monotonicity of Numerical Invariants for the Submodules $[(z-w)^k]$ in $H^2(\mathbb D^2)$

Abstract: For $k\geq1$, let $$M_k=[(z-w)^k]\subset H^2(\mathbb D<sup>2)$$. We first determine the banded Toeplitz matrices associated with the homogeneous components of $M_k$, together with explicit formulas for their determinants and the relevant algebraic cofactors. These formulas lead to a complete description of the spectrum of the core operator: [ σ(C_{M_k}) = {0,1} \cup \left{ \pm\frac{k}{n+k}:n\geq1 \right}. ] In particular, the spectral data determine the parameter $k$. The determinant and cofactor formulas further yield a unified finite-sum representation for $$α<em>{n,j}^{(k)} =\langle w<sup>jφ_n,z<sup>jψ_n\rangle$$, and hence for Yang's higher numerical invariants. We derive an adjacent relation connecting $α</em>{n,j}<sup>{(k)}$ and $α<em>{n,j+1}<sup>{(k)}$ by means of an explicit telescoping certificate, and show that the corresponding finite-section transformations are strict contractions. Combining these finite-dimensional estimates with the asymptotic behavior of $α</em>{n,j}<sup>{(k)}$, we prove the strict monotonicity [ Σ_0(M_k)> Σ_1(M_k)> Σ_2(M_k)> \cdots . ] The cases $k\geq3$ constitute the new part of the analysis, while the previously known cases $k=1,2$ are recovered within the same framework. Consequently, Yang's monotonicity conjecture holds in strict form for the entire family ${[(z-w)<sup>k]:k\geq1}$.

• The paper proves Yang’s conjectured strict monotonicity, showing that Σ₀(Mₖ)>Σ₁(Mₖ)>Σ₂(Mₖ)>⋯ for every k≥1, with Σ₀(Mₖ)=k²∑ₘ₌ₖ∞m⁻² and Σ₁(Mₖ)=Σ₀(Mₖ)−1.
• The authors combine banded Toeplitz Gram matrices, explicit determinant and cofactor formulas, and a telescoping adjacent relation to convert successive invariant levels into a strict finite-section contraction.
• The core spectrum is {0,1}∪{±k/(n+k): n≥1}, so it determines k and, together with Hilbert–Schmidt estimates, shows that every nonzero core-operator eigenvalue is simple.

Overview and main results

This paper studies the principal homogeneous submodules $M_k=[(z-w)^k]\subset H^2(\mathbb D^2)$ for $k\geq 1$, with the goal of verifying Yang's monotonicity conjecture for this family. The conjecture asserts that for any submodule $M\subset H^2(\mathbb D^2)$, the sequence of higher numerical invariants

$$\Sigma_j(M)=\sum_{n=0}^{\infty}\left|\langle w^j\phi_n,z^j\psi_n\rangle\right|^2,\qquad j\geq 0,$$

where $\{\phi_n\}$ and $\{\psi_n\}$ are homogeneous orthonormal bases of the defect spaces $M\ominus zM$ and $M\ominus wM$, is nonincreasing in $j$. The conjecture remains open in general, so explicit computations on concrete families serve as tests of the phenomenon.

The main theorem establishes strict monotonicity for all $k\geq 1$:

$k\geq 1$0

with closed forms $k\geq 1$1 and $k\geq 1$2. The cases $k\geq 1$3 were previously known; the new content is the uniform treatment of $k\geq 1$4.

Banded Toeplitz structure and the core spectrum

The repeated factor $k\geq 1$5 produces a fundamentally different matrix structure from polynomials with distinct linear factors such as $k\geq 1$6. The Gram matrices $k\geq 1$7 of the vectors $k\geq 1$8 are real symmetric $k\geq 1$9-banded Toeplitz matrices with entries determined by binomial coefficients:

$M\subset H^2(\mathbb D^2)$0

corresponding to the Fisher–Hartwig symbol $M\subset H^2(\mathbb D^2)$1. No residue-class decomposition is available, unlike the distinct-factor case.

Two combinatorial formulas drive the analysis. First, an inverse-entry formula for the last column,

$M\subset H^2(\mathbb D^2)$2

is proved by a finite-difference argument using a degree-$M\subset H^2(\mathbb D^2)$3 polynomial annihilated by $M\subset H^2(\mathbb D^2)$4; it can also be recovered from the Duduchava–Roch inversion formula. Second, the determinants satisfy

$M\subset H^2(\mathbb D^2)$5

and every last-row cofactor is positive, with opposite-corner cofactors $M\subset H^2(\mathbb D^2)$6.

These formulas yield a complete description of the core operator spectrum:

$M\subset H^2(\mathbb D^2)$7

with $M\subset H^2(\mathbb D^2)$8 of infinite multiplicity and $M\subset H^2(\mathbb D^2)$9 simple. A notable consequence: since the largest nontrivial positive eigenvalue is $$\Sigma_j(M)=\sum_{n=0}^{\infty}\left|\langle w^j\phi_n,z^j\psi_n\rangle\right|^2,\qquad j\geq 0,$$0, one has $$\Sigma_j(M)=\sum_{n=0}^{\infty}\left|\langle w^j\phi_n,z^j\psi_n\rangle\right|^2,\qquad j\geq 0,$$1, so the core spectrum alone determines the parameter $$\Sigma_j(M)=\sum_{n=0}^{\infty}\left|\langle w^j\phi_n,z^j\psi_n\rangle\right|^2,\qquad j\geq 0,$$2. This contrasts sharply with the family $$\Sigma_j(M)=\sum_{n=0}^{\infty}\left|\langle w^j\phi_n,z^j\psi_n\rangle\right|^2,\qquad j\geq 0,$$3, whose nonzero core spectrum is independent of $$\Sigma_j(M)=\sum_{n=0}^{\infty}\left|\langle w^j\phi_n,z^j\psi_n\rangle\right|^2,\qquad j\geq 0,$$4; there, factor multiplicity is invisible spectrally, whereas here it is fully encoded.

Finite-sum representation and the adjacent relation

The determinant and cofactor formulas reduce the double sum defining $$\Sigma_j(M)=\sum_{n=0}^{\infty}\left|\langle w^j\phi_n,z^j\psi_n\rangle\right|^2,\qquad j\geq 0,$$5 to a single finite sum via a Pfaff–Saalschütz $$\Sigma_j(M)=\sum_{n=0}^{\infty}\left|\langle w^j\phi_n,z^j\psi_n\rangle\right|^2,\qquad j\geq 0,$$6 evaluation. This gives an exact support result: $$\Sigma_j(M)=\sum_{n=0}^{\infty}\left|\langle w^j\phi_n,z^j\psi_n\rangle\right|^2,\qquad j\geq 0,$$7 for $$\Sigma_j(M)=\sum_{n=0}^{\infty}\left|\langle w^j\phi_n,z^j\psi_n\rangle\right|^2,\qquad j\geq 0,$$8, and the first nonzero coefficient in each column is explicitly known and nonzero.

The central structural identity is the adjacent relation

$$\Sigma_j(M)=\sum_{n=0}^{\infty}\left|\langle w^j\phi_n,z^j\psi_n\rangle\right|^2,\qquad j\geq 0,$$9

proved by an explicit telescoping certificate: each summand's contribution factors as a first difference $\{\phi_n\}$0 with vanishing endpoints. The proof handles boundary indices by rewriting everything without division, using rising-factorial products—a necessary technical device since boundary terms would otherwise be undefined.

Finite-section contractions and strict monotonicity

The adjacent relation recasts as $\{\phi_n\}$1 between finite coefficient vectors at levels $\{\phi_n\}$2 and $\{\phi_n\}$3, where $\{\phi_n\}$4 with $\{\phi_n\}$5 lower bidiagonal (hence invertible). Positivity of $\{\phi_n\}$6—established by Gershgorin diagonal dominance in the case $\{\phi_n\}$7, and by a positive diagonal scaling argument when $\{\phi_n\}$8—implies $\{\phi_n\}$9. Hence

$\{\psi_n\}$0

for every admissible $\{\psi_n\}$1. The paper emphasizes that this is a norm inequality, not a termwise one: individual inequalities $\{\psi_n\}$2 are false in general, and monotonicity arises from contraction of the whole vector rather than pointwise domination.

To preserve strictness in the limit, the asymptotic expansion

$\{\psi_n\}$3

shows that $\{\psi_n\}$4 eventually. Combining the positive finite-section differences with eventual termwise positivity yields $\{\psi_n\}$5 for all $\{\psi_n\}$6, and $\{\psi_n\}$7 follows from the corner-cofactor formula. As a corollary, matching the Hilbert–Schmidt norm against the spectral data shows that every nonzero eigenvalue of $\{\psi_n\}$8 is simple.

Limitations and open questions

Several qualifications apply. The analysis is specific to the repeated-factor structure of $\{\psi_n\}$9; the mechanism does not transfer to submodules with distinct linear factors, where block repetition rather than strict decrease occurs. The restriction to real coefficients suffices here but leaves the general complex-valued generator untouched. Most importantly, Yang's monotonicity conjecture remains open for general submodules of $M\ominus zM$0: this paper confirms it strictly only for the single family $M\ominus zM$1, and whether the adjacent-relation/contraction technique extends to other homogeneous principal submodules—or to finitely generated ones—is not addressed.

Conclusion

The paper provides a complete, uniform treatment of the numerical invariants for $M\ominus zM$2, combining explicit banded Toeplitz determinant and cofactor formulas, a full core-spectrum computation that detects the factor multiplicity $M\ominus zM$3, a telescoping-derived adjacent recurrence, and finite-section contraction estimates. The result verifies Yang's monotonicity conjecture in strict form for this family, with the cases $M\ominus zM$4 constituting the new contribution, and demonstrates that strict decrease is already visible at every finite graded section rather than emerging only from infinite-series cancellation.