Toral and spherical Aluthge transforms of $2$-variable weighted shifts

TL;DR

This paper introduces toral and spherical Aluthge transforms for 2-variable weighted shifts, explores their properties, and investigates the class of spherically quasinormal shifts as fixed points of the spherical transform.

Contribution

It defines and analyzes two new types of Aluthge transforms for 2-variable weighted shifts and characterizes spherically quasinormal shifts as their fixed points.

Findings

Spherical Aluthge transform fixed points are spherically quasinormal shifts.

Properties of toral and spherical Aluthge transforms are established.

Relation between spherically quasinormal and spherically isometric shifts is discussed.

Abstract

We introduce two natural notions of Aluthge transforms (toral and spherical) for 2-variable weighted shifts and study their basic properties. Next, we study the class of spherically quasinormal $2$-variable weighted shifts, which are the fixed points for the spherical Aluthge transform. Finally, we briefly discuss the relation between spherically quasinormal and spherically isometric 2-variable weighted shifts.