Square root problem and Subnormal Aluthge transforms

Abstract

For a non negative measure $\mu$ with $p$ atoms, we study the relation between the Square Root Problem of $\mu$ and the problem of subnormality of ${\tilde{W}}_{\mu }$ the Aluthge transform of the associated unilateral weighted shift. We use an approach based on uniquely represented elements in the support of $\mu \ast \mu$. We first show that if ${\tilde{W}}_{\mu }$ is subnormal, then $2p-1\le card(supp(\mu \ast \mu ))\le [\frac{(p-1{)}^2+6}{2}]$. We rewrite several results known for finitely atomic measure having at most five atoms and give a complete solution for measures six atoms.

19 pages, 1 figure

References in corpus (1)

• Berger measure for some transformations of subnormal weighted shifts

Cited by in corpus (1)

• The Square Root Problem and Subnormal Aluthge Transforms of Recursively Generated Weighted Shifts