Aluthge Transforms of 2-Variable Weighted Shifts

作者：Curto Raul E^*; Yoon Jasang

来源：Integral Equations and Operator Theory, 2018, 90(5): UNSP 52.

DOI：10.1007/s00020-018-2475-1

• Toral and spherical Aluthge transforms
• Hyponormal
• 2-Variable weighted shifts
• Taylor spectrum

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.

• 出版日期2018-10