Adjoints of Composition Operators on Hardy Spaces of the Half-Plane

TL;DR

This paper develops formulas for the adjoints of composition operators on Hardy spaces of the upper half-plane, extending known boundedness conditions and classifying bounded operators with rational symbols.

Contribution

It introduces new formulas for adjoints, extends boundedness criteria, and classifies bounded composition operators with rational symbols on Hardy spaces of the half-plane.

Findings

Derived explicit formulas for adjoints of composition operators

Extended boundedness criteria for these operators

Classified all bounded composition operators with rational symbols

Abstract

Building on techniques used in the case of the disc, we use a variety of methods to develop formulae for the adjoints of composition operators on Hardy spaces of the upper half-plane. In doing so, we prove a slight extension of a known necessary condition for the boundedness of such operators, and use it to provide a complete classification of the bounded composition operators with rational symbol. We then consider some specific examples, comparing our formulae with each other, and with other easily deduced formulae for simple cases.