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The Aluthge transform of unilateral weighted shifts and the Square Root Problem for finitely atomic measures
Author(s) -
Curto Raúl E.,
Kim Jaewoong,
Yoon Jasang
Publication year - 2019
Publication title -
mathematische nachrichten
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.913
H-Index - 50
eISSN - 1522-2616
pISSN - 0025-584X
DOI - 10.1002/mana.201800140
Subject(s) - mathematics , measure (data warehouse) , square root , probability measure , multiplicative function , borel measure , convolution (computer science) , interval (graph theory) , statement (logic) , square (algebra) , root (linguistics) , pure mathematics , combinatorics , discrete mathematics , mathematical analysis , geometry , database , machine learning , computer science , artificial neural network , linguistics , philosophy , political science , law
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.