On numerical invariants for submodules $[(z-w)^2]$ in $H^2(\mathbb{D}^2)$

Abstract: In this paper, we study numerical invariants associated with a homogeneous submodule of the Hardy module over the bidisk. We focus on the submodule generated by the polynomial $(z-w)<sup>2$ and obtain explicit formulas for the corresponding invariants. As an application, we verify the monotonicity property in this concrete setting. Our results provide a detailed example illustrating the behavior of these invariants beyond the linear case.

• The paper computes explicit Yang numerical invariants for [(z-w)²] in H²(𝔻²), validating the strict monotonicity of these metrics.
• It employs Toeplitz determinant methods and solution of difference equations to derive closed formulas and analyze the core operator spectrum.
• The study contrasts linear and quadratic generator cases, providing critical benchmarks for the operator-theoretic classification of polynomial-generated submodules.

Numerical Invariants for the Submodule $[(z-w)^2]$ in $H^2(\mathbb{D}^2)$

Introduction and Framework

The paper investigates numerical invariants associated with the submodule $[(z-w)^2]$ of the Hardy space $H^2(\mathbb{D}^2)$, viewed as a Hilbert module over $\mathbb{C}[z,w]$. The multidimensional Hardy module setting lacks a complete analog of the Beurling theorem, resulting in a rich and complex theory for submodules. Even explicit polynomial-generated submodules, such as those generated by $(z-w)^2$, are not unitarily equivalent to any module generated by a single inner function, and their structure must be analyzed operator-theoretically.

The approach centers on the analysis of Yang’s numerical invariants $\{\Sigma_k(M)\}_{k\geq 0}$, which quantify commutators of restricted multiplication operators. While explicit calculations of these invariants exist for $[z-w]$ and the full module, little was known for higher-degree homogeneous submodules. The work addresses this by undertaking a comprehensive computation for $[(z-w)^2]$ and rigorously verifying the monotonicity conjecture of the sequence $\{\Sigma_k\}$ in this concrete case.

Core Operator and Numerical Invariants

Let $H^2(\mathbb{D}^2)$0. The core operator $H^2(\mathbb{D}^2)$1 is defined via the core function $H^2(\mathbb{D}^2)$2, where $H^2(\mathbb{D}^2)$3 is the reproducing kernel for $H^2(\mathbb{D}^2)$4. The Hilbert-Schmidt norm and spectral structure of $H^2(\mathbb{D}^2)$5 encode substantial module-theoretic information.

Yang’s invariants $H^2(\mathbb{D}^2)$6 and $H^2(\mathbb{D}^2)$7 are defined as the squared Hilbert-Schmidt norms of certain commutators involving restricted multiplication operators $H^2(\mathbb{D}^2)$8 and $H^2(\mathbb{D}^2)$9. For homogeneous submodules, these invariants can be represented as squared inner products between elements in orthonormal bases of defect spaces $[(z-w)^2]$0 and $[(z-w)^2]$1. Higher invariants $[(z-w)^2]$2 generalize this to the action of powers $[(z-w)^2]$3.

A crucial technical ingredient is the direct link established between the structure of these spaces and the properties of explicit Toeplitz matrices and their determinants, associated with the symbol $[(z-w)^2]$4 for the generating homogeneous polynomial $[(z-w)^2]$5.

Explicit Computation for $[(z-w)^2]$6

Determinants and Cofactors

For $[(z-w)^2]$7, the relevant Toeplitz matrices are five-diagonal, with algebraically tractable recurrence relations. Calculations lead to closed formulas:

• $[(z-w)^2]$8
• $[(z-w)^2]$9

By leveraging results from explicit Toeplitz determinant theory ([BS], [Fisher]), these formulas enable direct computation of the invariants.

Base Numerical Invariants and Norms

Using established relations, the first two invariants are obtained: $H^2(\mathbb{D}^2)$0 with the Hilbert-Schmidt norm $H^2(\mathbb{D}^2)$1.

Higher-Order Invariants and Monotonicity

An elaborate decomposition of inner-products and a careful analysis of the associated Toeplitz matrix cofactors yield an explicit formula for $H^2(\mathbb{D}^2)$2: $H^2(\mathbb{D}^2)$3

The general case requires explicit calculation of the last row cofactors of the five-diagonal Toeplitz matrix, which is achieved by solving a third-order difference equation. This gives an explicit formula for all higher $H^2(\mathbb{D}^2)$4: $H^2(\mathbb{D}^2)$5 The asymptotic behavior is: $H^2(\mathbb{D}^2)$6 from which monotonicity and convergence to zero are immediate. The paper establishes that the sequence $H^2(\mathbb{D}^2)$7 is strictly decreasing for all $H^2(\mathbb{D}^2)$8, in accordance with the Yang conjecture for this explicit case.

Spectral Analysis

The core operator $H^2(\mathbb{D}^2)$9 associated with $\mathbb{C}[z,w]$0 has spectrum

$\mathbb{C}[z,w]$1

showing the decay rate of nontrivial eigenvalues and confirming the sharpness of lower bounds for the second largest eigenvalue achieved in previous estimates.

Comparison With the Linear Case $\mathbb{C}[z,w]$2

The analysis is extended to the linear generator $\mathbb{C}[z,w]$3, providing explicit cofactor and determinant structures. Here, the Toeplitz matrices are tridiagonal, with $\mathbb{C}[z,w]$4 and $\mathbb{C}[z,w]$5. The invariants for $\mathbb{C}[z,w]$6 are likewise computed explicitly, and it is shown that

$\mathbb{C}[z,w]$7

is strictly decreasing, with leading-order decay $\mathbb{C}[z,w]$8.

Implications and Future Directions

The rigorous, explicit computation of $\mathbb{C}[z,w]$9 for $(z-w)^2$0 provides a detailed benchmark for the monotonicity conjecture of Yang and demonstrates the tractability of these invariants for higher-degree homogeneous polynomial submodules. The methods, centered on precise Toeplitz determinant calculations and difference equations for cofactors, set the stage for similar analyses of other classes of polynomial-generated submodules.

These invariants quantify the deviation of the submodule structure from the trivial or Beurling-type case, and their decay encapsulates subtle spectral and operator-theoretic properties. The results have implications for the classification problem for polynomial-generated submodules, for understanding finer spectral data of associated compressed multiplication operators, and for potential connections with random matrix theory and multivariate operator theory.

The explicit methods may be adapted to more general homogeneous polynomials, although increased algebraic complexity is expected. The present results suggest that strict monotonicity of $(z-w)^2$1 may hold in wider generality, but a unified proof for all homogeneous (or more general) polynomial submodules remains open.

Conclusion

The paper provides a comprehensive and technically robust study of the numerical invariants for the submodule $(z-w)^2$2 in $(z-w)^2$3. Explicit formulas for invariants and core operator spectrum are derived, and the strict monotonicity of the sequence $(z-w)^2$4 is confirmed for this nontrivial example. These findings advance the operator-theoretic classification of Hardy module submodules and lay groundwork for further research on higher-degree and more general submodule structures.