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The Square Root Problem and Aluthge transforms of weighted shifts

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  • Sang Hoon Lee
  • Jasang Yoon

Abstract

In this paper we consider the following question. When does there exist a square root of a probability measure supported on R+? This question is naturally related to subnormality of weighted shifts. The main result of this paper is that if μ is a finitely atomic probability measure having at most 4 atoms, then μ has a square root, i.e., there exists a measure ν such that μ=ν*ν (* means the convolution) if and only if the Aluthge transform of a subnormal weighted shift with Berger measure μ is subnormal. As an application of them, we give non†trivial, large classes of probability measures having a square root. We also prove that there are 6 and 7†atomic probability measures which don't have any square root. Our results have a connection to the following long†open problem in Operator Theory: characterize the subnormal operators having a square root.

Suggested Citation

  • Sang Hoon Lee & Jasang Yoon, 2017. "The Square Root Problem and Aluthge transforms of weighted shifts," Mathematische Nachrichten, Wiley Blackwell, vol. 290(17-18), pages 2925-2933, December.
    • Handle: RePEc:bla:mathna:v:290:y:2017:i:17-18:p:2925-2933
    DOI: 10.1002/mana.201600302
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