tailieunhanh - Báo cáo toán học: "Positive semigroups on ordered Banach spaces "

Tuyển tập các báo cáo nghiên cứu khoa học ngành toán học tạp chí Journal of Operator Theory đề tài: Semigroups tích cực trên không gian Banach thứ tự. | J. OPERATOR THEORY 9 1983 371-400 g Copyright by INCREST 1983 POSITIVE SEMIGROUPS ON ORDERED BANACH SPACES OLA BRATTELI TROND DIGERNES and DEREK w. ROBINSON 0. INTRODUCTION Let be a Banach space ordered by a normal positive cone ăă with non-empty interior intổ . First we give a simple proof of a recent theorem of Arendt Chernoff and Kato H generates a positive cn-semigroup 5on if and only if I aZZ -1 is a bounded positive operator for all small a 0. Second we demonstrate that S intổ int and then relate a variety of stronger ergodic criteria of S . Next we examine c -semigroups. In this case it is not generally true that Sjnt int á but this condition is equivalent to D Zf n n intáZ . 7 0. Moreover H generates a positive Co -semigroup such that 5 intẩ if and only if I is a bounded positive operator for all small a 0 and D 77 f intổ 0. We conclude with some complementary remarks about semigroups on c - and JF -algebras. The aim of this paper is to characterize generators of positive co- and c -semi-groups . continuous semigroups on ordered Banach spaces ẵd which respect the order and to discuss various ergodicity properties of these semigroups. The infinitesimal characterization of continuous semigroups has been extensively studied for recent reviews see for example 3 5 6 . The classic Hille-Yosida theorem demonstrates that an operator H generates a c0-semigroup of contractions if and only if the resolvents Z xH -1 are well-defined contraction operators for all small a 0 and the Lumer-Phillips theorem re-expresses this property as a range condition R Z ẩỉ together with a condition of dissipativity of H. More generally the Feller-Miyadera-Phillips theorem establishes that bounds Z aZZ _ aco n 1 2 . are both necessary and sufficient for H to generate a general C0-semigroup. Some of these characterizations then extend to Of-semigroups by duality. The study of positive semigroups is much less complete. In 1962 Phillips 15 demonstrated that H generates a .