Yang's monotonicity conjecture for higher numerical invariants

Determine whether Yang's higher numerical invariant sequence \(\{\Sigma_j(M)\}_{j\geq0}\) is nonincreasing for every submodule \(M\) of the Hardy space \(H^2(\mathbb D^2)\).

Background

For a submodule M⊂H2(D2)M\subset H^2(\mathbb D^2), Yang's higher numerical invariants are defined by Σj(M)=∑n=0∞∣⟨wjϕn,zjψn⟩∣2\Sigma_j(M)=\sum_{n=0}^{\infty}|\langle w^j\phi_n,z^j\psi_n\rangle|^2, where {ϕn}\{\phi_n\} and {ψn}\{\psi_n\} are homogeneous orthonormal bases of the defect spaces M⊖zMM\ominus zM and M⊖wMM\ominus wM, respectively. Yang conjectured that these invariants form a nonincreasing sequence for every bidisk submodule.

The paper proves strict monotonicity for the specific family of principal homogeneous submodules Mk=[(z−w)k]M_k=[(z-w)^k], k≥1k\geq1. Thus, the conjecture remains unresolved only in its general form beyond the family treated in the paper.

References

Yang conjectured that the sequence ${\Sigma_j(M)}_{j\geq0}$ is nonincreasing for every submodule of $H2(\mathbb D2)$. The conjecture remains open in general, and explicit computations for concrete homogeneous submodules therefore provide useful tests of this operator-theoretic phenomenon .

— Strict Monotonicity of Numerical Invariants for the Submodules $[(z-w)^k]$ in $H^2(\mathbb D^2)$  (2608.17780 - Liu et al., 18 Aug 2026) in Section 1, Introduction, paragraph introducing Yang's higher numerical invariants

For

p_\theta(z,w)=z2-2\cos\theta\,zw+w2, \qquad 0<\theta<\pi,

determine the sign of

\Delta(\theta)=\Sigma_4([p_\theta])-\Sigma_3([p_\theta])

on the whole parameter interval. Determine all zeros of $\Delta$, and in particular decide the maximal interval containing $\pi/6$ on which $\Delta(\theta)>0$.

— Verblunsky coefficients, CMV matrices and numerical invariants of homogeneous bidisc submodules  (2608.18456 - Lu et al., 19 Aug 2026) in Problem 5, Section 10, "Problems and further directions"