Resolvent Positive Operators

Abstract

Resolvent positive operators on an ordered Banach space (with generating and normal positive cone) are by definition linear (possibly unbounded) operators whose resolvent exists and is positive on a right half-line. Even though these operators are defined by a simple (purely algebraic) condition, analogues of the basic results of the theory of C0-semigroups can be proved for them. In fact, if A is resolvent positive and has a dense domain, then the Cauchy problem associated with A has a unique solution for every initial value in the domain of A2, and the solution is positive if the initial value is positive. Also the converse is true (if we assume that A has a non-empty resolvent set and D(A2)∩E+ is dense in E+). Moreover, every positive resolvent is a Laplace–-Stieltjes transform of a so-called integrated semigroup; and conversely every such (increasing, non-degenerate) integrated semigroup defines a unique resolvent positive operator.