Generators of Feller semigroups as generators of \(L^p\)-sub-Markovian semigroups (Q2708589)

Let \((A^{(\infty)}, D(A^{(\infty)}))\) be the generator of a Feller semigroup \((T^{(\infty)}_t)_{t\geq 0}\) on \(C_\infty(\mathbb{R}^n; \mathbb{R}^1)\). It is proved that if there exists a dense subspace \(U\) of \(L^p(\mathbb{R}^n; \mathbb{R}^1)\), \(1< p< \infty\), such that \(A^{(\infty)}|U\) extends to a generator \(A^{(p)}\) of a strongly continuous contraction semigroup \((T^{(p)}_t)_{t\geq 0}\) on \(L^p(\mathbb{R}^n; \mathbb{R}^1)\) and \(V:= (\lambda- A^{(p)})^{- 1}U\), \(\lambda> 0\), is an operator core for \(A^{(p)}\), then \((A^{(p)}, D(A^{(p)}))\) is an \(L^p\)-Dirichlet operator, i.e. a closed, densely defined linear operator on \(L^p(\mathbb{R}^n; \mathbb{R}^1)\), and for all \(u\in D(A^{(p)})\) NEWLINE\[NEWLINE\int_{\mathbb{R}^n} (A^{(p)} u)(x)((u(x)- 1)^+)^{p- 1} dx\leq 0.NEWLINE\]NEWLINENEWLINENEWLINEFor the entire collection see [Zbl 0957.00037].