Continuity of the Norm of a Composition Operator

We explore the continuity of the map which, given an analytic selfmap of the disk, takes as its value the norm of the associated composition operator on the Hardy space H^(2) . We also examine the continuity the functions which assign to a self-map of the disk the Hilbert-Schmidt norm or the essential norm of the associated composition operator and show these to be discontinuous. Additionally, we characterize when the norm of a composition operator is minimal.