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Let Bn be the unit ball of ℂn and ϕ=(ϕ1,…,ϕn) a holomorphic self-map of Bn. Let 0-1, α>0, and let Cϕ be the composition operator between the space F(p,q,s) and α-Bloch space ℬα induced by ϕ. This paper gives an estimate of the essential norm of Cϕ. As a consequence, a necessary and sufficient condition for the composition operator Cϕ to be compact from F(p,q,s) to ℬα is obtained.

We characterize the isometries on the products of composition and integral operators on the Bloch type space in the disk.

Let BN be the unit ball in the N-dimensional complex space, for È, a holomorphic function in BN, and Õ, a holomorphic map from BN into itself, the weighted composition operator on the weighted Hardy space H2(β,BN) is given by (CÈ,Õ)f=È(z)f(Õ(z)), where f∈H2(β,BN). This paper discusses the spectrum of CÈ,Õ when it is compact on a certain class of weighted Hardy spaces and when the composition map Õ has only one fixed point inside the unit ball.