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 (T12, T2) may fail to be in H1.
Conversely, we find a pair (T1, T2)∈ H0 such that (T12, T2)∈ H1 but (T1, T2)∉ H1. Next, we
show that there exists a pair (T1, T2)∈ H1 such that T1mT2n is subnormal (for all m, n⩾ 1) …
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 (T12, T2) may fail to be in H1.
Conversely, we find a pair (T1, T2)∈ H0 such that (T12, T2)∈ H1 but (T1, T2)∉ H1. Next, we
show that there exists a pair (T1, T2)∈ H1 such that T1mT2n is subnormal (for all m, n⩾ 1) …
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 (T12,T2) may fail to be in H1. Conversely, we find a pair (T1,T2)∈H0 such that (T12,T2)∈H1 but (T1,T2)∉H1. Next, we show that there exists a pair (T1,T2)∈H1 such that T1mT2n 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 (T12,T2) and (T1,T22) does imply the subnormality of (T1,T2).
Elsevier
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