Hyponormality and subnormality for powers of commuting pairs of subnormal operators

Authors

• Raúl E. Curto, University of Iowa
• Sang Hoon Lee, University of Iowa
• Jasang Yoon, The University of Texas Rio Grande ValleyFollow

Document Type

Article

Publication Date

4-15-2007

Abstract

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 T1^m T2ⁿ 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). © 2007 Elsevier Inc. All rights reserved.

Comments

http://www.elsevier.com/open-access/userlicense/1.0/

Recommended Citation

Curto, Raúl E., Sang Hoon Lee, and Jasang Yoon. "Hyponormality and subnormality for powers of commuting pairs of subnormal operators." Journal of Functional Analysis 245, no. 2 (2007): 390-412. https://doi.org/10.1016/j.jfa.2007.01.002

First Page

390

Last Page

412

Publication Title

Journal of Functional Analysis

DOI

10.1016/j.jfa.2007.01.002