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

For 2-variable weighted shifts W_{(α,β)}(T_1, T_2) we study the invariance of (joint) k- hyponormality under the action (h,\ell) -> W_{(α,β)}^{(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_{(α,β)}(T_1, T_2) such that W_{(α,β)}^{(h,\ell)}(T_1, T_2) is k-hyponormal (all h>=2,\ell>=1) but W_{(α,β)}(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.

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