The Aluthge transform of unilateral weighted shifts and the Square Root Problem for finitely atomic measures

In this paper we consider the following Square Root Problem for measures: Given a positive probability Borel measure μ (supported on an interval [a,b]⊆R+), does there exist a positive Borel measure ν such that μ=ν*ν holds? (Here * denotes the multiplicative convolution, properly defined on R+.) This problem is intimately connected to the subnormality of the Aluthge transform of a unilateral weighted shift. We develop a criterion to test whether a measure μ admits a square root, and we provide a concrete solution for the case of a finitely atomic measure having at most five atoms. In addition, we sharpen the statement of a previous result on this topic and extend its applicability via a new technique that uses the standard inequality of real numbers to generate a diagram of a partial order on the support of a probability measure.