Tuyển tập các báo cáo nghiên cứu khoa học ngành toán học tạp chí Journal of Operator Theory đề tài: Một khái quát của dissipativity và semigroups tích cực. | J. OPERATOR THEORY 8 1982 167-180 Copyright by INCREST 1982 A GENERALIZATION OF DISSIPATIVITY AND POSITIVE SEMIGROUPS WOLFGANG ARENDT PAUL R. CHERNOFF and TOSIO KATO INTRODUCTION The starting point of this paper is a half-norm Ị on a Banach space X that is a positive homogeneous subadditive real-valued function on X. Such a half-norm defines a positive cone x x d - x 0 and we assume in addition that this cone is proper. Conversely every closed proper cone x in a Banach space is generated by a continuous half-norm . p x -- distf X X Ỵ . In Part I we develop a theory of -contraction semigroups and the associated cl ass of i -dissipative operators. These semigroups are in particular positive . they leave the cone x invariant. If 0 x x l p-dissipativity is simply dissipativity and we recover the Hille-Yosida theorem X is trivial in this case . If p x ịx X being a Banach lattice 0-dissipativity is the same as dispersiveness as introduced by Phillips 15 . As has been done in these special cases -dissipativity can be expressed in terms of the subdifferential d of . We also give a notion of strict d -dissipativity and there is a remarkable result cP-dissipativity implies strict 0-dissipativity if the operator is densely defined. In Part II we consider an ordered Banach space X whose positive cone x has non-empty interior. Every u e int Y defines in a natural way a half-norm p which generates the given cone. Applying the results of Part 1 to these half-norms we show that if the cone is normal a densely defined operator A in X is the infinitesimal generator of a C0-semigroup if and only .if its resolvent R Ấ A exists and is positive for all large real Ẳ. The latter property in turn can be expressed by the usual range conditions together with a minimum principle P which has been considered by Evans and Hanche-Olsen 6 for bounded generators. We conclude with an application of our general theory to the case when X is the space of hermitian elements of a G -algebra where .
