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.

Let γ (m) ≡ {γ ij } 0≤i+j≤m be a given complex-valued sequence. The truncated complex moment problem (TCMP in short) involves determining necessary and sufficient conditions for the existence of a positive Borel measure µ on C (called a representing measure for γ (m) ) such that γ ij = z i z j dµ for 0 ≤ i + j ≤ m. The TCMP has been completely solved only when m = 1, 2, 3, 4.We provide in this paper a concrete solution to the quintic TCMP (that is, when m = 5). We also study the cardinality of the minimal representing measure. Based on the bivariate recurrences sequences's properties with some Curto-Fialkow's results, our method intended to be useful for all odd-degree moment problems.2010 Mathematics Subject Classification. Primary 47A57, 15A83; Secondary 30E05, 44A60.

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