When is hyponormality for 2-variable weighted shifts invariant under powers?

TL;DR

This paper investigates when hyponormality of 2-variable weighted shifts remains unchanged under taking powers, revealing conditions and classes where invariance holds or fails, and connecting hyponormality to subnormality.

Contribution

It provides new criteria and examples for the invariance of k-hyponormality under powers of 2-variable weighted shifts, including classes with tensor cores and explicit determinant formulas.

Findings

Existence of shifts where powers are k-hyponormal but original is not.

A necessary condition for invariance in shifts with tensor core.

Identification of classes where hyponormality invariance implies subnormality.

Abstract

For 2-variable weighted shifts W_{(\alpha,\beta)}(T_1, T_2) we study the invariance of (joint) k- hyponormality under the action (h,\ell) -> W_{(\alpha,\beta)}^{(h,\ell)}(T_1, T_2):=(T_1^k,T_2^{\ell}) (h,\ell >=1). We show that for every k >= 1 there exists W_{(\alpha,\beta)}(T_1, T_2) such that W_{(\alpha,\beta)}^{(h,\ell)}(T_1, T_2) is k-hyponormal (all h>=2,\ell>=1) but W_{(\alpha,\beta)}(T_1, T_2) is not k-hyponormal. On the positive side, for a class of 2-variable weighted shifts with tensor core we find a computable necessary condition for invariance. Next, we exhibit a large nontrivial class for which hyponormality is indeed invariant under all powers; moreover, for this class 2-hyponormality automatically implies subnormality. Our results partially depend on new formulas for the determinant of generalized Hilbert matrices and on criteria for their positive semi-definiteness.