Journal of Operator Theory

Volume 64, Issue 1, Summer 2010 pp. 131-147.

On honesty of perturbed substochastic

$C_0$ -semigroups in

$L^1$ -spaces

Authors: Mustapha Mokhtar-Kharroubi (1) and Juergen Voigt (2)
Author institution: (1) Departement de Mathematiques, Universite de Franche-Comte, 16 Route de Gray, F-25030 Besancon, France
(2) Fachrichtung Mathematik, Technische Universit\"at Dresden, D-01062 Dresden, Germany

Summary: Let

$T$ be the generator of a positive contraction semigroup on

$L^{1}(\Omega ,\mu )$ , and let

$B\colon D(T)\rightarrow L^{1}(\Omega ,\mu )$ be a positive linear operator such that

$\int(Tf+Bf)\leq 0$ for all

$f\in D(T)_{+}$ . It is known that there exists a minimal positive contraction semigroup generated by some operator

$K\supseteq T+B.$ This paper deals mainly with the total mass carried by trajectories

$( \eul^{tK}f;t\geq 0 )$ with non-negative initial data

$f$ . In particular, our analysis covers the problem of whether

$K=\overline{T+B}$ or

$K\supsetneq \overline{T+B}$ \ and the related problem of the stochasticity or lack of stochasticity of `formally conservative'' perturbed positive semigroups in

$L^{1}$ -spaces.

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