Quasinormality of powers of commuting pairs of bounded operators

Authors

Raul E. Curto, University of Iowa
Sang H. Lee
Jasang Yoon, The University of Texas Rio Grande ValleyFollow

Document Type

Article

Publication Date

2-2020

Abstract

We study jointly quasinormal and spherically quasinormal pairs of commuting operators on Hilbert space, as well as their powers. We first prove that, up to a constant multiple, the only jointly quasinormal 2-variable weighted shift is the Helton-Howe shift. Second, we show that a left invertible subnormal operator T whose square T2 is quasinormal must be quasinormal. Third, we generalize a characterization of quasinormality for subnormal operators in terms of their normal extensions to the case of commuting subnormal n-tuples. Fourth, we show that if a 2-variable weighted shift W(α,β) and its powers W(α,β)(2,1) and W(α,β)(1,2) are all spherically quasinormal, then W(α,β) may not necessarily be jointly quasinormal. Moreover, it is possible for both W(α,β)(2,1) and W(α,β)(1,2) to be spherically quasinormal without W(α,β) being spherically quasinormal. Finally, we prove that, for 2-variable weighted shifts, the common fixed points of the toral and spherical Aluthge transforms are jointly quasinormal.

Comments

Original published version available at https://doi.org/10.1016/j.jfa.2019.108342

Recommended Citation

Curto, Raúl E., Sang Hoon Lee, and Jasang Yoon. "Quasinormality of powers of commuting pairs of bounded operators." Journal of Functional Analysis 278, no. 3 (2020): 108342. https://doi.org/10.1016/j.jfa.2019.108342

Creative Commons License

This work is licensed under a Creative Commons Attribution-NonCommercial-No Derivative Works 4.0 International License.

Publication Title

Journal of Functional Analysis

DOI

10.1016/j.jfa.2019.108342