tailieunhanh - Báo cáo toán học: "On the smoothness of sphere extensions "

Tuyển tập các báo cáo nghiên cứu khoa học ngành toán học tạp chí Journal of Operator Theory đề tài: Trên suốt các phần mở rộng mặt cầu. | Copyright by INCREST 1981 J. OPERATOR THẸORỴ 6 1981 I 13 123 AN ABSTRACT KATO INEQUALITY FOR GENERATORS OF POSITIVE OPERATORS SEMIGROUPS ON BANACH LATTICES REINER NAGEL and HEINRICH UHLIG 1. THE CONJECTURE The main theme in the theory of strongly continuous semigroups of linear operators on Banach spaces is the interplay between three objects i the semigroup T t l 0 which is a subset of L E E a Banach space such that t T t is a continuous semigroup homomorphism for the strong operator topology on L E . ii the generator A D A which is a generally unbounded linear operator with dense domain Ũ A in E defined by . lim T t X .v T . l- 0 t iii the resolvent R z A which is a bounded linear operator defined as the inverse of z A for large positive ze R. The famous Hille-Yosida theorem characterizes those unbounded linear operators A jD A which are generators of such semigroups T t 0 but its conditions rely heavily on properties of the resolvent R Z A see 18 Ch. IX . On the other hand the Lumer-Phillips theorem succeeds in characterizing the generator more directly again see 18 but unfortunately this theory works only for contraction semigroups. Due to the fact that many semigroups appearing in the applications . partial differential equations probability theory are semigroups of positive operators on function spaces bearing a natural order structure making those spaces into Banach lattices analogous problems arise. Using the Hille-Yosida theorem and the relations TO lim t t I 8 - Ỉ242 114 REINER NAGEL and HEINRICH UHLIG and R z A c e_JU T s di Jo in the strong sense one can easily characterize the generator of positive semigroups in terms and by the positivity of the resolvent. A direct characterization of such generators is desirable but more difficult to obtain. One of the first results in this direction used in the theory of Markov processes and analogous to the Lumer-Phillips theorem was found for positive contraction semigroups on spaces Cc X X .