  Suppose $X$ is a right process which is associated with a non-symmetric
Dirichlet form $(\mathcal{E},D(\mathcal{E}))$ on $L^{2}(E;m)$. For $u\in
D(\mathcal{E})$, we have Fukushima's decomposition:
$\tilde{u}(X_{t})-\tilde{u}(X_{0})=M^{u}_{t}+N^{u}_{t}$. In this paper, we
investigate the strong continuity of the generalized Feynman-Kac semigroup
defined by $P^{u}_{t}f(x)=E_{x}[e^{N^{u}_{t}}f(X_{t})]$. Let
$Q^{u}(f,g)=\mathcal{E}(f,g)+\mathcal{E}(u,fg)$ for $f,g\in
D(\mathcal{E})_{b}$. Denote by $J_1$ the dissymmetric part of the jumping
measure $J$ of $(\mathcal{E},D(\mathcal{E}))$. Under the assumption that $J_1$
is finite, we show that $(Q^{u},D(\mathcal{E})_{b})$ is lower semi-bounded if
and only if there exists a constant $\alpha_0\ge 0$ such that
$\|P^{u}_{t}\|_2\leq e^{\alpha_0 t}$ for every $t>0$. If one of these
conditions holds, then $(P^{u}_{t})_{t\geq0}$ is strongly continuous on
$L^{2}(E;m)$. If $X$ is equipped with a differential structure, then this
result also holds without assuming that $J_1$ is finite.
