On Shapiro's compactness criterion for composition operators

We give an elementary and direct proof of the identity; lim sup vertical bar(w vertical bar -> 1-) N(psi) (W)/vertical bar 1-vertical bar W vertical bar = lim sup(vertical bar a vertical bar -> 1-)(1 - vertical bar a vertical bar(2)) parallel to 1/(1- (a) over bar psi)parallel to(2)(H2) for any analytic self-map psi of {z: vertical bar z vertical bar < 1}: where N(psi) denotes the Nevanlinna counting function of psi. We further show that one can find analytic self-maps psi of {z: vertical bar z vertical bar < 1}, where the composition operator C(psi) on the Hardy space H(2) is compact, such that parallel to psi(n)parallel to(H2) tends to zero at an arbitrarily slow rate, as n -> infinity; even in the case that psi is univalent. Among these are new examples, where C(psi) is compact on H(2), but not in any of the Schatten classes. (C) 2010 Elsevier Inc. All rights reserved.