Premium
Resolvent Positive Operators
Author(s) -
Arendt Wolfgang
Publication year - 1987
Publication title -
proceedings of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.899
H-Index - 65
eISSN - 1460-244X
pISSN - 0024-6115
DOI - 10.1112/plms/s3-54.2.321
Subject(s) - mathematics , resolvent , resolvent formalism , pure mathematics , converse , semigroup , banach space , domain (mathematical analysis) , operator (biology) , mathematical analysis , finite rank operator , biochemistry , chemistry , geometry , repressor , transcription factor , gene
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 C 0 ‐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 A 2 , 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 ( A 2 )∩ 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.