Hilbert Transformation and Representation of the ax + b Group

Abstract

AbstractIn this paper we study the Hilbert transformations over L2() and L2() fromthe viewpoint of symmetry. For a linear operator over L2() commutative with the ax + b group, we show that the operator is of the form λI+ηH, where I and H are the identity operator and Hilbert transformation, respectively, and λ, η 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 L2().