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arXiv · 2105.10612

Square root problem and Subnormal Aluthge transforms

Hamza El Azhar·Abdelouahab Hanine·Kaissar Idrissi·El Hassan Zerouali
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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.

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Hamza El Azhar, Abdelouahab Hanine, Kaissar Idrissi, El Hassan Zerouali. 2021-05-22. Square root problem and Subnormal Aluthge transforms. https://doi.org/10.1007/s43034-022-00232-2

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