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).