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 $\mu *\mu$. We first show that if ${{\tilde{W}}_\mu }$ is subnormal, then $2p-1\le card(supp(\mu *\mu ))\le [\frac{(p-1)^2+6}{2}]$. We rewrite several results known for finitely atomic measure having at most five atoms and give a complete solution for measures six atoms.