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Strict Monotonicity of Numerical Invariants for the Submodules [(z-w)^k] in H²( D²) · arXivDesk

2608.17780Aug 18, 202627 pages

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

Yin Liu, Yufeng Lu, Chao Zu

Functional Analysis Complex Variables

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Abstract

For $k\geq1$ , let $M_k=[(z-w)^k]\subset H^2(\mathbb D^2)$

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. 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_{n,j}^{(k)} =\langle w^jφ_n,z^jψ_n\rangle

, and hence for Yang's higher numerical invariants. We derive an adjacent relation connecting

α_{n,j}^{(k)}

and

α_{n,j+1}^{(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_{n,j}^{(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)^k]:k\geq1\}

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