双圆盘Hardy空间中子模$[z^k-w^k]$的数值不变量的块重复
Block Repetition of Numerical Invariants for the Submodules $[z^k-w^k]$ in $H^2(\mathbb D^2)$
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中文总结 AI 辅助
该研究确定双圆盘Hardy空间主齐次子模$M_k=[z^k-w^k]$的Yang数值不变量序列，证明Yang单调性猜想成立，发现核心算子非零谱数据与$k$无关，高阶数值不变量可检测核心谱无法察觉的模论信息。
中文摘要 AI 辅助
对于$k\ge2$，令$M_k=[z^k-w^k]$为双圆盘Hardy空间的主齐次子模。我们确定了Yang的完整数值不变量序列，并证明：$\Sigma_0(M_k)=\frac{\pi^2}{6}$，$\Sigma_j(M_k)=\Sigma_{\lceil j/k\rceil}([z-w])$（$j\ge1$）。证明利用了与分级游荡空间相关的Toeplitz矩阵的剩余类分解。由此得出，Yang的单调性猜想对族$\{M_k:k\ge2\}$成立。我们还证明了核心算子的非零谱数据与$k$无关，而数值不变量序列可通过其常数块的长度恢复$k$，因此高阶数值不变量能检测核心谱无法察觉的模论信息。
英文摘要
For $k\ge 2$, let $M_k=[z^k-w^k]$ be the principal homogeneous submodule of the Hardy space over the bidisk. We determine Yang's complete sequence of numerical invariants and prove $$ Σ_0(M_k)=\frac{π^2}{6},\qquad Σ_j(M_k)=Σ_{\lceil j/k\rceil}([z-w]),\quad j\ge1. $$ The proof exploits a residue-class decomposition of the Toeplitz matrices associated with the graded wandering spaces. Consequently, Yang's monotonicity conjecture holds for the family $\{M_k:k\ge2\}$. We also show that the nonzero spectral data of the core operator are independent of $k$, whereas the numerical invariant sequence recovers $k$ from the length of its constant blocks. Thus the higher numerical invariants detect module-theoretic information invisible to the core spectrum.
