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Báo cáo toán học: "Positive semigroups and resolvents "

Tuyển tập các báo cáo nghiên cứu khoa học ngành toán học tạp chí Journal of Operator Theory đề tài: Tích cực semigroups và chỉ nhưng thuốc làm tiêu độc. | J. OPERATOR THEORY 10 1983 357-363 Copyright by INCREST 1983 POSITIVE SEMIGROUPS AND RESOLVENTS c. J. K. BATTY and E. B. DAVIES 1. INTRODUCTION The genera theory of Co-semigroups s st t 0 of operators on Banach spaces is well established by now see 5 10 13 but in most applications there is some additional structure present which often allows some reformulation and simplification of their properties. It has recently been proved 1 that the fundamental condition 11 2-Z - II M 2-a - for the operator z on a real Banach space 3 to be the generator of a Co-semigroup can be replaced by positivity of the resolvent if ỖS is ordered by a normal generating positive cone ẩ assuming that has non-empty interior interior points of are the order-units for 3 . It is shown in Theorem 1 below that this last assumption cannot be omitted and in Theorems 2 and 3 that other conjectures made in 4 are false. For positive Cfl-semigroups s on several different types of ordered Banach space it has been established that the limit cos lim t l log S i oo belongs to the spectrum of the generator. In Theorem 4 this will be shown to be the case if 38 is also a-directed for example if Să is the self-adjoint part of any c -algebra. Finally in Theorem 5 it is shown that cos may be determined by inspection of a single interior point of and extremal functionals in 3S . 2. GENERATORS AND POSITIVE RESOLVENTS Throughout the paper Ỗ8 will denote a real Banach space ordered by a normal generating cone ỖỈ . It was shown in 1 see also 4 that if ổ has non--empty interior then a densely-defined linear operator z on á is the generator of 358 c. J. K. BATTY and E. B. DAVIES a positive co-semigroup if and only if the resolvents R 2 Z 2Z Z 1 exist and are positive operators on 29 for all sufficiently large real 2. It was conjectured in 4 that this result remains true even if 29 has no interior but the following theorem shows that the result is in fact false for 29 C0 R the space of continuous real functions on R .