Constructions of Rota-Baxter operators by L-R smash products

Let $A$ and $H$ be two cocommutative Hopf algebras such that $A$ is an $H$-bimodule Hopf algebra. Suppose that $R:A\rightarrow A$ is a linear map and $B$ is a Rota-Baxter operator of $H$. In this paper we will characterize the Rota-Baxter operators on the L-R smash product $A\natural H$ and give the necessary and sufficient conditions to make $\overline{B}$ a Rota-Baxter operator of $A\natural H$. Then we will consider the dual case, and construct a Rota-Baxter co-operator on the L-R smash coproduct $C\ltimes H$, where $C$ and $H$ are commutative Hopf algebras and $C$ is an $H$-bicomodule Hopf algebra.