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 (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 T m 1 T n 2 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 (T 2 1 ,T2) and (T1,T 2 2 ) does imply the subnormality of (T1,T2). © 2007 Elsevier Inc. All rights reserved