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Resolvent positive operators. (English) Zbl 0617.47029
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. 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.

47D03(Semi)groups of linear operators
47B60Operators on ordered spaces
46B42Banach lattices
44A10Laplace transform
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