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Hilbert Transformation and Representation of the ax + b Group

Published online by Cambridge University Press:  20 November 2018

Pei Dang ,
Hua Liu  and
Tao Qian
Pei Dang
Affiliation:
Faculty of Information Technology, Macau University of Science and Technology, Macau, China, e-mail: pdang@must.edu.mo
Hua Liu
Affiliation:
Department of Mathematics, Tianjin University of Technology and Education, Tianjin 300222, China, e-mail: daliuhua@163.com
Tao Qian
Affiliation:
Department of Mathematics, University of Macau, Macau, China, e-mail: fsttq@umac.mo
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Abstract

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In this paper we study the Hilbert transformations over ${{L}^{2}}(\mathbb{R})$ and ${{L}^{2}}(\mathbb{T})$ from the viewpoint of symmetry. For a linear operator over ${{L}^{2}}(\mathbb{R})$ commutative with the $ax\,+\,b$ group, we show that the operator is of the form $\lambda I+\eta H$, where $I$ and $H$ are the identity operator and Hilbert transformation, respectively, and $\lambda ,\eta $ are complex numbers. In the related literature this result was proved by first invoking the boundedness result of the operator using some machinery. In our setting the boundedness is a consequence of the boundedness of the Hilbert transformation. The methodology that we use is the Gelfand–Naimark representation of the $ax\,+\,b$ group. Furthermore, we prove a similar result on the unit circle. Although there does not exist a group like the $ax\,+\,b$ group on the unit circle, we construct a semigroup that plays the same symmetry role for the Hilbert transformations over the circle ${{L}^{2}}(\mathbb{T})$.

Keywords

30E2544A1542A50singular integralHilbert transformthe ax + b group
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 61 , Issue 1 , 01 March 2018 , pp. 70 - 84
Copyright
Copyright © Canadian Mathematical Society 2018

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