Abstract:
In joint work with Jasang Yoon, we study the invariance of the Taylor joint spectrum under the spherical Aluthge transform for commuting pairs of Hilbert space operators.  We pay special attention to the class of 2-variable weighted shifts, including the Drury-Arveson 2-shift.  We will first review the basic properties of the spherical Aluthge transform, and how it applies to the case of 2-variable weighted shifts.  We will then consider whether subnormality is preserved under this transform.  Next, we will discuss the class of spherically quasinormal 2-variable weighted shifts, which are the fixed points of the spherical Aluthge transform.  Finally, we will see how recursive relations in the 0-th row of a quasinormal 2-variable weighted shift propagate throughout the entire shift, leading up to a finitely atomic Berger measure for the shift.