AI 中文总结
该研究针对齐次双圆盘子模，利用Verblunsky系数、CMV矩阵等构造加权模型，推导奇异值、谱等数值不变量，给出二次反例反驳不变量单调性。
AI 中文摘要
设$[p]$为$H^2(\boldsymbol{\text{D}}^2)$中由多项式生成的主子模。对于齐次多项式$p$，$[p]$的齐次切片允许一个加权OPUC模型，其中两个游荡向量为正交多项式及其反转多项式。我们证明，相关的Verblunsky系数决定了游荡投影乘积、受限交叉交换子的奇异值，以及核心算子的非零谱。Toeplitz行列式与Mahler测度恒等式给出了精确的Fredholm行列式与Schatten估计，而$[(z-w)^N]$排除了一致Hilbert-Schmidt界。该模型还给出了齐次商空间$H^2(\boldsymbol{\text{D}}^2)\backslash[p]$上$[S_z^*,S_w]$的显式奇异值；当$p=(z-w)^N$时，其平方Hilbert-Schmidt范数渐近于$N$。对于任意多项式生成元，我们构造了一个带双重Toeplitz、块带矩矩阵的加权二元模型，并证明$C_p^2|_{\boldsymbol{\text{E}}_z}=\boldsymbol{\text{Γ}}_p^*\boldsymbol{\text{Γ}}_p$，将核心谱与两个边缘空间间的交叉Gram算子关联起来。我们还讨论了循环因子障碍，通过交替CMV乘积表示更高阶数值不变量，并给出了一个二次反例以反驳其提出的单调性。
英文摘要
Let $[p]$ be the principal submodule generated by a polynomial in $H^2(\mathbb D^2)$. 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 $[(z-w)^N]$ rules out a uniform Hilbert--Schmidt bound. The same model gives explicit singular values of $[S_z^*,S_w]$ on the homogeneous quotient $H^2(\mathbb D^2)\ominus[p]$; for $p=(z-w)^N$, 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 $C_p^2|_{\mathscr E_z}=Γ_p^*Γ_p$, 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.
Comments35 pages
