  Let $k:E\times E\to [0,\infty)$ be a non-negative measurable function on some
locally compact separable metric space $E$. We provide some simple conditions
such that the quadratic form with jump kernel $k$ becomes a regular lower
bounded (non-local, non-symmetric) semi-Dirichlet form. If $E=\R^n$ we identify
the generator of the semi-Dirichlet form and its (formal) adjoint. In
particular, we obtain a closed expression of the adjoint of the stable-like
generator $-(-\Delta)^{\alpha(x)}$ in the sense of Bass. Our results complement
a recent paper by Fukushima and Uemura (2012) and establishes the relation of
these results with the symmetric principal value (SPV) approach due to Zhi-ming
Ma and co-authors (2006).
