A note on absorption semigroups and regularity
type: Item
instance of: scholarly article
title: A note on absorption semigroups and regularity
zbMATH Open document ID: 1361.47015
author: Amir Manavi
author: Hendrik Vogt
author: Jürgen Voigt
published in: Archiv der Mathematik
publication date: 2016-05-17T00:00:00Z
full work available at URL: https://arxiv.org/abs/1508.03182
review text: 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.
reviewed by: Hector O. Fattorini
Mathematics Subject Classification ID: 47D06
Mathematics Subject Classification ID: 47A55
Mathematics Subject Classification ID: 47B60
zbMATH DE Number: 6581299
zbMATH Keywords: semigroup perturbation
zbMATH Keywords: absorption semigroup
zbMATH Keywords: regular absorption rate
MaRDI profile type: Publication
OpenAlex ID: W3101735586
arXiv ID: 1508.03182
cites work: Absorption semigroups and Dirichlet boundary conditions
cites work: Admissible and regular potentials for positive \(C_ 0\)-semigroups and application to heat semigroups
cites work: Bands in Lp-spaces
cites work: Absorption semigroups, their generators, and Schrödinger semigroups
cites work: Q3480762
DOI: 10.1007/S00013-016-0892-4
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description: scientific article; zbMATH DE number 6581299