Quasinormality of powers of commuting pairs of bounded operators

TL;DR

This paper investigates the properties of commuting operator pairs on Hilbert space, establishing conditions under which their powers are quasinormal or spherically quasinormal, and characterizing their joint quasinormality and fixed points of certain transforms.

Contribution

It provides new characterizations of quasinormality for operator pairs, including the Helton-Howe shift uniqueness, and explores the relationships between powers and joint quasinormality.

Findings

The only jointly quasinormal 2-variable weighted shift (up to a constant) is the Helton-Howe shift.

A left invertible subnormal operator with a quasinormal square must be quasinormal.

Fixed points of toral and spherical Aluthge transforms are jointly quasinormal.

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 $T^{2}$ 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 $W_{\left(\alpha ,\beta \right) }$ and its powers $W_{\left(\alpha ,\beta \right)}^{(2,1)}$ and $W_{\left(\alpha ,\beta \right)}^{(1,2)}$ are all spherically quasinormal, then $W_{\left( \alpha ,\beta \right)}$ may not necessarily be jointly quasinormal. Moreover, it is possible for both $W_{\left(\alpha ,\beta \right)}^{(2,1)}$ and $W_{\left(\alpha ,\beta \right)}^{(1,2)}$ to be spherically quasinormal without $W_{\left(\alpha ,\beta \right)}$ 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.