Hyponormality and subnormality for powers of commuting pairs of subnormal operators

Details Hyponormality and subnormality for powers of commuting pairs of subnormal operators Hyponormality and subnormality for powers of commuting pairs of subnormal operators

2006-10-28

arxiv.org/abs/math/0610888v1

J. Funct. Anal. 245(2007), 390-412.

Mathematics

Functional Analysis

Scientific paper

Let H_0 (resp. H_\infty denote the class of commuting pairs of subnormal operators on Hilbert space (resp. subnormal pairs), and for an integer k>=1 let H_k denote the class of k-hyponormal pairs in H_0. We study the hyponormality and subnormality of powers of pairs in H_k. We first show that if (T_1,T_2) is in H_1, then the pair (T_1^2,T_2) may fail to be in H_1. Conversely, we find a pair (T_1,T_2) in H_0 such that (T_1^2,T_2) is in H_1 but (T_1,T_2) is not. Next, we show that there exists a pair (T_1,T_2) in H_1 such that T_1^mT_2^n is subnormal (all m,n >= 1), but (T_1,T_2) is not in H_\infty; this further stretches the gap between the classes H_1 and H_\infty. 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) 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).

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