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Resolvent positive operators. (English) Zbl 0617.47029

Let A be a resolvent positive (linear) operator (i.e., (λ-A) -1 exists and is positive for λ>λ 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 0 -semigroups.

For example, if D(A) is dense or E is reflexive, then for every xD(A 2 ) there exists a unique classical solution of the abstract Cauchy problem

(ACP)u(t)=Au(t)(t0),U(0)=x

and u(t)0 (t0) if x0. Moreover, A is the generator of a so- called integrated semigroup; i.e. there exists S: [0,)L(E) strongly continuous s.t. (λ-A) -1 =λ 0 e -λt S(t)dt (λ>λ 0 ). The solution of (ACP) is given by

u(t)=S(t)Ax+x·

A variety of examples is given.


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