On Shapiroʼs compactness criterion for composition operators

We give an elementary and direct proof of the identity: lim sup | w | → 1 − N ψ ( w ) 1 − | w | = lim sup | a | → 1 − ( 1 − | a | 2 ) ‖ 1 / ( 1 − a ¯ ψ ) ‖ H 2 2 , for any analytic self-map ψ of { z : | z | < 1 } ; where N ψ 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 C ψ on the Hardy space H 2 is compact, such that ‖ ψ n ‖ H 2 tends to zero at an arbitrarily slow rate, as n → ∞ ; even in the case that ψ is univalent. Among these are new examples, where C ψ is compact on H 2 , but not in any of the Schatten classes.