Square root problem and Subnormal Aluthge transforms

TL;DR

This paper explores the relationship between the Square Root Problem of a measure with atoms and the subnormality of its Aluthge transform, providing new bounds and solutions for measures with up to six atoms.

Contribution

It establishes bounds on the support of the convolution of measures and offers a complete solution for measures with six atoms, advancing understanding of subnormality conditions.

Findings

If the Aluthge transform is subnormal, then support size bounds are established.

Reformulation of known results for measures with up to five atoms.

Complete solution provided for measures with six atoms.

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.