Essential Norms of Weighted Composition Operators between Hardy Spaces in the unit Ball

الأصل · EN

Let ϕ(z)=(ϕ₁(z),...,ϕₙ(z)) be a holomorphic self-map of Bₙ and ψ(z) a holomorphic function on Bₙ, and H(Bₙ) the class of all holomorphic functions on Bₙ, where Bₙ is the unit ball of Cⁿ, the weight composition operator Wψ,ϕ is defined by Wψ,ϕ=ψf(ϕ) for f∈ H(Bₙ). In this paper we estimate the essential norm for the weighted composition operator Wψ,ϕ acting from the Hardy space Hᵖ to Hq (0<p,q≤ ∞). When p=∞ and q=2, we give an exact formula for the essential norm. As their applications, we also obtain some sufficient and necessary conditions for the bounded weighted composition operator to be compact from Hᵖ to Hq.