Hilbert Transformation and Representation of ax+b Group

TL;DR

This paper characterizes operators commuting with the ax+b group as linear combinations of the identity and Hilbert transform, using Gelfand-Naimark representation, and extends the analysis to the unit circle with a constructed semigroup.

Contribution

It provides a new proof of the form of operators commuting with the ax+b group using symmetry and representation theory, and extends the framework to the circle with a semigroup.

Findings

Operators commuting with ax+b are of the form λI + ηH.

The boundedness of such operators follows from the boundedness of the Hilbert transform.

A semigroup on the circle mimics the ax+b group symmetry for Hilbert transforms.

Abstract

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 through first invoking the boundedness result of the operator, proved though a big machinery. In our setting the boundedness is a consequence of the boundedness of the Hilbert transformation. The methodology that we use is Gelfand-Naimark's representation of the ax+b group. Furthermore we prove a similar result on the unit circle. Although there does not exist a group like ax+b on the unit circle, we construct a semigroup to play the same symmetry role for the Hilbert transformations over the circle $L^2(\mathbb{T}).$