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

Raúl E. Curto, Jaewoong Kim and Jasang Yoon

Mathematische Nachrichten, 2019, vol. 292, issue 11, 2352-2368

Abstract: 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.

Date: 2019
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