Aluthge transforms of 2-variable weighted 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.

References in corpus (3)

• Propagation phenomena for hyponormal 2-variable weighted shifts
• Reconstruction of the Berger measure when the core is of tensor form
• Spectral pictures of 2-variable weighted shifts

Cited by in corpus (5)

• Quasinormality of powers of commuting pairs of bounded operators
• The Spectral Picture and Joint Spectral Radius of the Generalized Spherical Aluthge Transform
• Joint spectra of spherical Aluthge transforms of commuting n-tuples of Hilbert space operators
• Joint numerical radius of spherical Aluthge transforms of tuples of Hilbert space operators
• Semi-hyponormality of commuting pairs of Hilbert space operators