Square root problem and subnormal Aluthge transforms

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