Symmetries and Reflections from Composition Operators in the Disk

Abstract

The set Q of reflections (i.e., operators C such that C2 = I) ina C∗ -algebra is a geometric space which has been the object of several investigations, and is an important tool in the study of these algebras. In this paper we consider a special class of reflections, the compositionoperators Ca acting on the Hardy space H2 of the unit disk, given by C a f = f o φa, where φa(z) =a − z1 − ¯az , for |a| < 1. These operators are indeed reflections, because φa o φa = id. We study their eigenspaces N(Ca ± I), their relative position (i.e., the intersections between these spaces and their orthogonal complements for a ≠ b in the unit disk) and the symmetries induced by Ca and these eigenspaces.