Hyponormality and subnormality for powers of commuting pairs of subnormal operators

Let h(0) (respectively h(infinity)) denote the class of commuting pairs of subnormal operators on Hilbert space (respectively subnormal pairs), and for an integer k >= 1 let S)k denote the class of k-hyponormal pairs in h(0). We study the hyponormality and subnormality of powers of pairs in hl(k). We first show that if (T-1, T-2) c S) 1, the pair (T-1(2) T-2) may fail to be in h(1). Conversely, we find a pair (T-1, T-2) is an element of h(0) such that (T2, T2) E F) I but (T-1, T-2) is not an element of h(1). Next, we show that there exists a pair (T-1, T-2) is an element of h(1) such that T-1(m) T-2(n) is subnormal (for all in, n >= 1), but (T-1, T-2) is not in S)cc; this further stretches the gap between the classes h(1) and h(infinity). Finally, we prove that there exists a large class of 2-variable weighted shifts (T-1, T-2) (namely those pairs in h(0) whose cores are of tensor form (cf. Definition 3.4)), for which the subnormality of (T-1(2), T-2) and (T-1, T-2())2 does imply the subnormality of (T-1, T-2). (c) 2007 Elsevier Inc. All rights reserved.