School of Mathematical and Statistical Sciences Faculty Publications and Presentations

The Square Root Problem and Aluthge transforms of weighted shifts

Document Type

Article

Publication Date

12-2017

Abstract

In this paper we consider the following question. When does there exist a square root of a probability measure supported on ? 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 (* 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.

Comments

© 2017 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim.

https://onlinelibrary.wiley.com/share/GX2FTBHXWBG5K54IZF6X?target=10.1002/mana.201600302

Publication Title

Mathematische Nachrichten

DOI

10.1002/mana.201600302

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