Suppose that X is a right process which is associated with a non-symmetric Dirichlet form \documentclass[12pt]{minimal}
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\begin{document}$(\mathcal{E},D(\mathcal{E}))$\end{document} on L2(E;m). For \documentclass[12pt]{minimal}
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\begin{document}$u\in D(\mathcal{E})$\end{document}, we have Fukushima’s decomposition: \documentclass[12pt]{minimal}
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\begin{document}$\tilde{u}(X_{t})-\tilde{u}(X_{0})=M^{u}_{t}+N^{u}_{t}$\end{document}. In this paper, we investigate the strong continuity of the generalized Feynman–Kac semigroup defined by \documentclass[12pt]{minimal}
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\begin{document}$P^{u}_{t}f(x)=E_{x}[e^{N^{u}_{t}}f(X_{t})]$\end{document}. Let \documentclass[12pt]{minimal}
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\begin{document}$Q^{u}(f,g)=\mathcal{E}(f,g)+\mathcal{E}(u,fg)$\end{document} for \documentclass[12pt]{minimal}
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\begin{document}$f,g\in D(\mathcal{E})_{b}$\end{document}. Denote by J1 the dissymmetric part of the jumping measure J of \documentclass[12pt]{minimal}
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\begin{document}$(\mathcal{E},D(\mathcal{E}))$\end{document}. Under the assumption that J1 is finite, we show that \documentclass[12pt]{minimal}
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\begin{document}$(Q^{u},D(\mathcal{E})_{b})$\end{document} is lower semi-bounded if and only if there exists a constant α0≥0 such that \documentclass[12pt]{minimal}
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\begin{document}$\|P^{u}_{t}\|_{2}\leq e^{\alpha_{0}t}$\end{document} for every t>0. If one of these conditions holds, then \documentclass[12pt]{minimal}
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\begin{document}$(P^{u}_{t})_{t\geq0}$\end{document} is strongly continuous on L2(E;m). If X is equipped with a differential structure, then this result also holds without assuming that J1 is finite.
