# Aluthge transforms of 2-variable weighted shifts

## TL;DR

This paper introduces and analyzes two types of multivariable Aluthge transforms for 2-variable weighted shifts, revealing their effects on hyponormality, spectral properties, and fixed points, with implications for multivariable operator theory.

## Contribution

It defines toral and spherical Aluthge transforms for multivariable shifts and studies their properties, including hyponormality preservation and spectral behavior, which were previously unexplored.

## Key findings

- Toral Aluthge transform does not preserve hyponormality in 2-variable shifts.
- A large class of 2-variable shifts retains hyponormality under both transforms.
- Spectral properties are affected by the transforms, with fixed points being spherically quasinormal shifts.

## Abstract

We introduce two natural notions of multivariable Aluthge transforms (toral and spherical), and study their basic properties. In the case of 2-variable weighted shifts, we first prove that the toral Aluthge transform does not preserve (joint) hyponormality, in sharp contrast with the 1-variable case. Second, we identify a large class of 2-variable weighted shifts for which hyponormality is preserved under both transforms. Third, we consider whether these Aluthge transforms are norm-continuous. Fourth, we study how the Taylor and Taylor essential spectra of 2-variable weighted shifts behave under the toral and spherical Aluthge transforms; as a special case, we consider the Aluthge transforms of the Drury-Arveson 2-shift. Finally, we briefly discuss the class of spherically quasinormal 2-variable weighted shifts, which are the fixed points for the spherical Aluthge transform.