Essential norm of weighted composition operators on Bargmann-Fock spaces

Abstract

Let ϕ be an entire self-map of the n-dimensional Euclidean complex space Cn and ψ be an entire function on Cn. A weighted composition operator induced by ϕ with weight ψ is given by (Wψ,ϕf)(z) = ψ(z)f(ϕ(z)), for z∈Cn and f is entire function on Cn. In this paper, we study weighted composition operators between Bargmann-Fock spaces Fpα(Cn) and Fpα(Cn) for 0 < p,q < ∞. Using Carleson-type measures techniques, we characterize the boundedness and compactness of these operators, when 0 < p,q < ∞. We also obtained an estimate of the essential norm of these operators, when 1 < p ≤ q < ∞. The results written in terms of a certain Berezin-type integral transform on Cn.