I have a question about convergence of resolvents of Markov processes.

Let $X$, $X^n$ be Markov processes on a locally compact separable metric space $E$. We denote $\{ R_{\alpha}\}_{\alpha>0}$ and $\{ R_{\alpha}^n\}_{\alpha>0}$ by the resolvents of $X$ and $X^n$, respectively.

We assume the following:

• for any bounded measurable function $f$ and $\alpha>0$, $R_{\alpha}f$ and $R_{\alpha}^nf$ are continuous functions on $E$.
• for any bounded measurable function $f$ and $\alpha>0$, $\lim_{n \to \infty}\sup_{x \in E}|R_{\alpha}f(x)-R_{\alpha}^nf(x)|=0$.

My question

Let $\{p_t\}_{t>0}$ and $\{p_t^n\}$ be the semigroups of $X$ and $X^n$, respectively. Then, can we have the following?

• for any bounded measurable function $f$ and $\alpha>0$, $\lim_{n \to \infty}\sup_{x \in E}|p_tf(x)-p_t^nf(x)|=0$.

There exists a related result in Theorem 3.4.2 of Pagy's book. However we cannot apply this results to my question since the space of continuous functions on $E$ is not a Banach space. If you know any other related results, please let me know.