COMPACT COMPOSITION OPERATORS ON THE HARDY-ORLICZ AND WEIGHTED BERGMAN-ORLICZ SPACES ON THE BALL

Using recent characterizations of the compactness of composition operators on the Hardy-Orlicz and Bergman-Orlicz spaces on the ball [3], [4], we first show that a composition operator which is compact on every Hardy-Orlicz (or Bergman-Orlicz) space has to be compact on H-infinity. Then, although it is well-known that a map whose range is contained in some nice Koranyi approach region induces a compact composition operator on H-p (B-N) or on A(alpha)(p) (B-N), we prove that, for each Koranyi region Gamma, there exists a map phi : B-N -> Gamma such that C-phi is not compact on H-psi (B-N), when psi grows fast. Finally, we extend (and simplify the proof of) a result by K. Zhu for the classical weighted Bergman spaces, by showing that, under reasonable conditions, a composition operator C-phi is compact on the weighted Bergman-Orlicz space A(alpha)(psi)(B-N), if and only if lim vertical bar z vertical bar -> 1 psi(-1)(1/(1 -vertical bar phi(z)vertical bar)(N(alpha)))/psi(-1)(1/(1 - vertical bar z vertical bar)(N(alpha))) = 0. In particular, we deduce that the compactness of composition operators on A(alpha)(psi)(B-N) does not depend on alpha anymore when the Orlicz function psi grows fast.