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Zbl 0617.47029
Arendt, Wolfgang
Resolvent positive operators.
(English)
[J] Proc. Lond. Math. Soc., III. Ser. 54, 321-349 (1987). ISSN 0024-6115; ISSN 1460-244X/e

Let A be a resolvent positive (linear) operator (i.e., $(\lambda -A)\sp{- 1}$ exists and is positive for $\lambda >\lambda\sb 0)$ 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 $C\sb 0$-semigroups. \par For example, if D(A) is dense or E is reflexive, then for every $x\in D(A\sp 2)$ there exists a unique classical solution of the abstract Cauchy problem $$(ACP)\quad u(t)=Au(t)\quad (t\ge 0),\quad U(0)=x$$ and u(t)$\ge 0$ (t$\ge 0)$ if $x\ge 0$. Moreover, A is the generator of a so- called integrated semigroup; i.e. there exists S: [0,$\infty)\to L(E)$ strongly continuous s.t. $(\lambda -A)\sp{-1}=\lambda \int\sp{\infty}\sb{0}e\sp{-\lambda t}S(t)dt$ $(\lambda >\lambda\sb 0)$. The solution of (ACP) is given by $$u(t)=S(t)Ax+x.$$ A variety of examples is given.
MSC 2000:
*47D03 (Semi)groups of linear operators
47B60 Operators on ordered spaces
46B42 Banach lattices
44A10 Laplace transform

Keywords: completely monotonic; resolvent positive (linear) operator; Banach lattice; positive $C\sb 0$-semigroups; abstract Cauchy problem; generator; integrated semigroup

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