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

Abstract

For $k\ge 1$, 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 ${\alpha }_{n,j}^{(k)}=⟨w^j{\varphi }_n,z^j{\psi }_n⟩$, and hence for Yang's higher numerical invariants. We derive an adjacent relation connecting ${\alpha }_{n,j}^{(k)}$ and ${\alpha }_{n,j+1}^{(k)}$ 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 ${\alpha }_{n,j}^{(k)}$, we prove the strict monotonicity \[ Σ_0(M_k)> Σ_1(M_k)> Σ_2(M_k)> \cdots . \] The cases $k\ge 3$ 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 $\{[(z-w{)}^k]:k\ge 1\}$.

27 pages