Let \(\mathcal{E}\) be an ordered Banach space with additive norm on the positive cone \(\mathcal{E}_+\) (an ‘abstract state space’). Let \(\Phi\) be a positive linear functional on \(\mathcal{E}\) satisfying \(\langle \Phi, u\rangle =||u||\) for \(u\in \mathcal{E}_+\).
Let \(\left(U(t)\right)_{t\geq 0}\) be a substochastic \(C_0\)-semigroup on \(\mathcal{E}\), let \(\left(\mathcal{A}, \mathcal{D(A)}\right)\) denote the generator, and let \(\mathcal{B}:\mathcal{D(B)}\to \mathcal{E}\) be a non-negative operator with \(\mathcal{D(B)}\supseteq\mathcal{D(A)}\), satisfying the dissipativity condition \(\langle \Phi, (\mathcal{A}+\mathcal{B})u\rangle \leq 0\) for \(u\in\mathcal{D(B)}\cap \mathcal{E}_+\). According to the approach of T. Kato [J. Math. Soc. Japan 6, 1–15 (1954; Zbl 0058.10701)], it follows that there exists an extension \(\mathcal{K}\) of the sum \(\mathcal{A}+\mathcal{B}\) generating a substochastic semigroup \(\left(\mathcal{V}(t)\right)_{t\geq 0}\) which is representable by a (strongly convergent) Dyson-Phillips series representation.
In the ‘formally conservative case’, i.e., \(\langle \Phi, (\mathcal{A}+\mathcal{B})u\rangle = 0\) for \(u\in\mathcal{D(B)}\cap \mathcal{E}_+\), \(\left(\mathcal{V}(t)\right)_{t\geq 0}\) is called ‘honest’ if \(\mathcal{K} = \overline{\mathcal{A}+\mathcal{B}}\); this is the case iff \(||\mathcal{V}(t)u|| =||u||\) for all \(u\in \mathcal{E}_+\) and \(t\geq 0\).
If \(\mathcal{K}\) is a proper extension of \(\mathcal{A}+\mathcal{B}\), the definition of honesty is more involved, indeed too technical to be repeated in a review.
In the paper under review, the authors extend the theory of addition of generators and of honest semigroups to ‘evolution families’ \(\left(U(s,t)\right)_{s\leq t}\) perturbated by a measurable one-parameter family of non-negative operators \(\{\mathcal{B}(t): t\geq 0\}\), to obtain (under suitable conditions) a substochastic evolution family \(\left(V(s,t)\right)_{s\leq t}\), representable by a (modified) Dyson-Phillips series representation. Essential for the following is the well-known fact that evolution families \(\left(U(s,t)\right)_{s\leq t}\) resp. \(\left(V(s,t)\right)_{s\leq t}\) correspond to substochastic \(C_0\)-semigroups \(\left(\mathcal{T}_0(t)\right)_{t\geq 0}\) and \(\left(\mathcal{T}(t)\right)_{t\geq 0}\), respectively defined on the enlarged state space \(\mathcal{X}:=L^1(\mathbb{R}_+, \mathcal{E})\). The further investigations rely on the construction of a non-negative operator \(\widehat{\mathcal{B}}\) on \(\mathcal{X}\) such that the generator of \(\left(\mathcal{T}(t)\right)\) is an extension of the sum \(\mathcal{Z}+ \widehat{\mathcal{B}}\), \(\mathcal{Z}\) denoting the generator of the un-perturbed semigroup \(\left(\mathcal{T}_0(t)\right)\). (Technical properties are postponed to the appendix.) Thus, roughly spoken, the theory of honest evolution families (on \(\mathcal{E}\)) can be reduced to the theory of honest \(C_0\)-semigroups (on \(\mathcal{X}\)).
The theory is applied in Section 5 to linear Boltzmann equations (satisfying additional conditions as the ‘sub-critical hypothesis’ in neutron transport theory).
It could be mentioned that substochastic evolution families and perturbations are also investigated in probabilities on groups in connection with ‘convolution hemigroups’ (or distributions of additive processes) and the corresponding convolution operators. However, at least up to now, the theory of ‘honest’ hemigroups seems to be not relevant in this setup.