Hyponormality and subnormality for powers of commuting pairs of subnormal operators

Let H₀ (respectively H_∞) denote the class of commuting pairs of subnormal operators on Hilbert space (respectively subnormal pairs), and for an integer k?1 let Hk denote the class of k-hyponormal pairs in H₀. We study the hyponormality and subnormality of powers of pairs in Hk. We first show that if (T₁,T₂)∈H₁, the pair may fail to be in H₁. Conversely, we find a pair (T₁,T₂)∈H₀ such that but (T₁,T₂)∉H₁. Next, we show that there exists a pair (T₁,T₂)∈H₁ such that is subnormal (for all m,n?1), but (T₁,T₂) is not in H_∞; this further stretches the gap between the classes H₁ and H_∞. Finally, we prove that there exists a large class of 2-variable weighted shifts (T₁,T₂) (namely those pairs in H₀ whose cores are of tensor form (cf. Definition 3.4)), for which the subnormality of and does imply the subnormality of (T₁,T₂).