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Lower bounded semi-Dirichlet forms associated with Lévy type operators

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 $-(-Δ)^{α(x)}$ 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).