Weighted Composition Operators from H∞ to the Bloch Space on the Polydisc

Songxiao Li; Stević, Stevo
January 2007
Abstract & Applied Analysis;2007, p1
Academic Journal
Let 픻n be the unit polydisc of ℂn, ϕ(z) = (ϕ1(z),…,ϕn(z)) be a holomorphic self-map of 픻n, and ψ(z) a holomorphic function on 픻n. Let H(픻n) denote the space of all holomorphic functions with domain 픻n, H∞(픻n) the space of all bounded holomorphic functions on 픻n, and ℬ(픻n) the Bloch space, that is, ℬ(픻n) = {f ϵ H(픻n)∣ ∥ f ∥ℬ = ∣ f (0)∣+supzε픻n Σk=1n ∣(∂ f /∂zk)(z)∣(1- ∣zk∣²) < +∞}. We give necessary and sufficient conditions for the weighted composition operator ψCϕ induced by ϕ(z) and ψ(z) to be bounded and compact from H∞(픻n) to the Bloch space ℬ(픻n).


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