A construction of Rota-Baxter operators on cocommutative Hopf algebras

Let H and K be two cocommutative Hopf algebras over a field F such that (H, center dot) is a left K-module Hopf algebra. Denote by H#K the smash product Hopf algebra of H and K. Let R be an F-linear map from H to H, and B a Rota-Baxter operator on K. We prove that the linear map (B) over bar from H#K to H#K defined as (B) over bar (h#k) = B(k(1)) center dot R(h)#B(k(2)), for all h is an element of H and k is an element of K, is a Rota-Baxter operator on H#K if and only if R is a Rota-Baxter operator on H satisfying two suitable equations.