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 Wµ 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 Wµ is subnormal, then 2pWe rewrite several results known for finitely atomic measure having at most five atoms and give a complete solution for measures six atoms.