Toral and Spherical Aluthge Transforms

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 discuss the class of spherically quasinormal $2$-variable weighted shifts, which are the fixed points for the spherical Aluthge transform. The talk is based on joint work with Jasang Yoon.