Composition operators induced by symbols defined on a polydisk

Suppose ϕ is a holomorphic mapping from the polydisk Dm into the polydisk Dn, or from the polydisk Dm into the unit ball Bn, we consider the action of the associated composition operator Cϕ on Hardy and weighted Bergman spaces of Dn or Bn.We first find the optimal range spaces and then characterize compactness. As a special case, we show that if ϕ(z) = ϕ1(z), . . . , ϕn(z) , z= (z1, . . . , zn), is a holomorphic self-map of the polydisk Dn, then Cϕ maps A pα (Dn) boundedly into A p β (Dn), the weight β = n(2+ α) −2 is best possible, and the operator Cϕ :A pα Dn →A p β Dn is compact if and only if the function n k=1 1 − |zk|2 n n k=1 1− ϕk(z) 2 tends to 0 as z approaches the full boundary of Dn. This settles an outstanding problem concerning composition operators on the polydisk. © 2005 Elsevier Inc. All rights reserved