On Shapiroʼs compactness criterion for composition operators

We give an elementary and direct proof of the identity:lim sup|w|→1−Nψ(w)1−|w|=lim sup|a|→1−(1−|a|2)‖1/(1−a¯ψ)‖H22, for any analytic self-map ψ of {z:|z|<1}; where Nψ denotes the Nevanlinna counting function of ψ. We further show that one can find analytic self-maps ψ of {z:|z|<1}, where the composition operator Cψ on the Hardy space H2 is compact, such that ‖ψn‖H2 tends to zero at an arbitrarily slow rate, as n→∞; even in the case that ψ is univalent. Among these are new examples, where Cψ is compact on H2, but not in any of the Schatten classes.