When Is Hyponormality for 2-variable Weighted Shifts Invariant under Powers?

Abstract. For 2-variable weighted shifts W(α,β) ≡ (T1, T2) we study the invariance of (joint) khyponormality under the action (h, `) 7→ W (h,`) (α,β) := (T h 1 , T ` 2 ) (h, ` ≥ 1). We show that for every k ≥ 1 there exists W(α,β) such that W (h,`) (α,β) is k-hyponormal (all h ≥ 2, ` ≥ 1) but W(α,β) 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. Finally, we show that there exists a 2-hyponormal W(α,β) such that W (2,1) (α,β) is not 2-hyponormal. Our results partially depend on new formulas for the determinant of generalized Hilbert matrices and on criteria for their positive semi-definiteness.