Adjoints of composition operators on Hardy spaces of the half-plane

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. © 2008 Elsevier Inc. All rights reserved