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Resolvent positive operators. (English) Zbl 0617.47029

Let A be a resolvent positive (linear) operator (i.e., ${\left(\lambda -A\right)}^{-1}$ exists and is positive for $\lambda >{\lambda }_{0}\right)$ on a Banach lattice E. Even though no norm condition on the resolvent is demanded, a theory is developed which - to a large extent - is analogous to the theory of positive ${C}_{0}$-semigroups.

For example, if D(A) is dense or E is reflexive, then for every $x\in D\left({A}^{2}\right)$ there exists a unique classical solution of the abstract Cauchy problem

$\left(ACP\right)\phantom{\rule{1.em}{0ex}}u\left(t\right)=Au\left(t\right)\phantom{\rule{1.em}{0ex}}\left(t\ge 0\right),\phantom{\rule{1.em}{0ex}}U\left(0\right)=x$

and u(t)$\ge 0$ (t$\ge 0\right)$ if $x\ge 0$. Moreover, A is the generator of a so- called integrated semigroup; i.e. there exists S: [0,$\infty \right)\to L\left(E\right)$ strongly continuous s.t. ${\left(\lambda -A\right)}^{-1}=\lambda {\int }_{0}^{\infty }{e}^{-\lambda t}S\left(t\right)dt$ $\left(\lambda >{\lambda }_{0}\right)$. The solution of (ACP) is given by

$u\left(t\right)=S\left(t\right)Ax+x·$

A variety of examples is given.

##### MSC:
 47D03 (Semi)groups of linear operators 47B60 Operators on ordered spaces 46B42 Banach lattices 44A10 Laplace transform