AI 中文总结
本文研究H²(D²)中模[(z-w)^k]的数值不变量，通过推导相关矩阵公式、算子谱及相邻关系，证明Yang的单调性猜想对k≥1的整个族成立严格形式。
AI 中文摘要
对于k≥1，设M_k=[(z-w)^k]⊂H²(D²)。我们首先确定与M_k的齐次分量相关的带状托普利茨矩阵，以及它们行列式和相关代数余子式的显式公式。这些公式完整描述了核心算子的谱：σ(C_{M_k})={0,1}∪{±k/(n+k):n≥1}，谱数据可确定参数k。行列式和余子式公式进一步得到α_{n,j}^{(k)}=⟨w^jφ_n,z^jψ_n>的统一有限和表示，进而得到Yang的高阶数值不变量。我们通过显式望远镜证书推导了连接α_{n,j}^{(k)}和α_{n,j+1}^{(k)}的相邻关系，证明对应的有限截面变换是严格压缩映射。结合这些有限维估计和α_{n,j}^{(k)}的渐近行为，我们证明严格单调性Σ₀(M_k)>Σ₁(M_k)>Σ₂(M_k)>…。k≥3的情况是分析的新部分，而先前已知的k=1,2的情况可在同一框架内导出。因此，Yang的单调性猜想对整个族{[(z-w)^k]:k≥1}成立严格形式。
英文摘要
For $k\geq1$, let $M_k=[(z-w)^k]\subset H^2(\mathbb D^2)$. 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 $α_{n,j}^{(k)} =\langle w^jϕ_n,z^jψ_n\rangle$, 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 $\{[(z-w)^k]:k\geq1\}$.
Comments27 pages
