Weighted composition operators on Bergman spaces Aωp$A^p_\omega$

Abstract

Let 𝜙 be an analytic self-map of the open unit disk 𝔻, and let 𝑢 be an analytic function on 𝔻. The weighted composition operator induced by 𝜙 with weight 𝑢 is given by (𝑢𝐶 𝜙 𝑓)(𝑧) = 𝑢(𝑧)𝑓(𝜙(𝑧)) for 𝑧 in 𝔻 and 𝑓 analytic on 𝔻. In this paper, we study weighted composition operators acting between two exponentially weighted Bergman spaces 𝐴 𝑝 𝜔 and 𝐴 𝑞 𝜔 . We characterize the bounded, compact and Schatten class membership operators 𝑢𝐶 𝜙 acting from 𝐴 𝑝 𝜔 to 𝐴 𝑞 𝜔 when 0 < 𝑝 ≤ ∞ and 0 < 𝑞 < ∞. To obtain this, we first get an important estimate for the norm of the reproducing kernel in 𝐴 𝑝 𝜔 and some new characterizations of Carleson measures. Our results use certain integral transforms that generalize the usual Berezin transform. In the case where 𝑝 = 𝑞 and 𝑢 = 1, we compare our criteria with those given by Kriete and MacCluer in [15].