A note on absorption semigroups and regularity
Let \((\Omega, {\mathcal A}, \mu)\) be a measure space, \(1 \leq p < \infty\) and \(T(t)\) a positive \(C_0\)-semigroup in \(L^p(\mu)\) with infinitesimal generator \(A.\) The subject of this paper is to associate a \(C_0\)-semigroup with the formal expression \(A - V\) as generator, where \(V: \Omega \to [-\infty, \infty]\) is the \textit{absorption rate}. If \(V: \Omega \to [-\infty, 0]\) or \(V: \Omega \to [0, \infty]\), then \(V\) is called \(T\)-\textit{admissible} if it is locally measurable and the \textit{absorption semigroup} \[ T_V (t) = \text{s-}\hskip -0.8em\lim_{ n \to -\infty} e^{t(A - (V \land n) \lor (-n)) } \] exists for all \(t \geq 0\) with \(T_V (t)\) a \(C_0\)-semigroup [\textit{J. Voigt}, J. Funct. Anal. 67, 167--205 (1986; Zbl 0628.47027)]. Extra regularity of \(V\) results in extra properties of the absorption semigroup. One notion of regularity allows the computation of the generator for \(p = 1\) while another (seemingly stronger) produces dominated convergence theorems for absorption semigroups. The authors show that these two versions of regularity are in fact equivalent.
scientific article; zbMATH DE number 6581299