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

Abstract

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 $${\Sigma }_0(M_k)=\frac{{\pi }^2}{6},{\Sigma }_j(M_k)={\Sigma }_{\lceil j/k\rceil }([z-w]),j\ge 1.$$ 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 $\{M_k:k\ge 2\}$. 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.

17 pages