Square root problem and Subnormal Aluthge transforms

Abstract

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