Hyponormality and Subnormality for Powers of Commuting Pairs of Subnormal Operators

Let H0 (resp. H∞) denote the class of commuting pairs of subnormal operators on Hilbert space (resp. 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 (T 2 1 , T2) may fail to be in H1. Conversely, we find a pair (T1, T2) ∈ H0 such that (T 2 1 , T2) ∈ H1 but (T1, T2) / ∈ H1. Next, we show that there exists a pair (T1, T2) ∈ H1 such that Tm 1 T n 2 is subnormal (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) ∈ H0, i.e., those whose core is of tensor form (cf. Definition 3.3), for which the subnormality of (T 2 1 , T2) and (T1, T 2 2 ) does imply the subnormality of (T1, T2).