Characterization of the generators of $C_0$ semigroups which leave a convex set invariant
H. N. Bojadziev · Czech digital mathematics library · 1984
Ue consider the problem: Given a Banach space X,a closed convex, subset K € X uith nonempty interior and a C D semigroup Tt (i i 0) on X uith generator A,find necessary and sufficient conditions for A so that T^K £ K for every t $£ O.To obtain a characterization of such generators ue introduce tuo boundary principles uhich are generalizations of the minimum principles used in f1] ,£33,1.8.1 to characterize the generators of positive C 0 semigroups on some ordered Banach spaces.Kay uords: Convex set,tangent functional,linear operator, dissipative operator, C Q semigroup,order unit space,positive semigroup • Classification; 47B44,47B55,47005,47H20. I.Introduction.Uhen given a C semigroup T. (t = 0) on a Banach space X,an important problem is to connect the properties of T. uith those of its generator A.It is uell-knoun,for example, that T. is a contraction semigroup iff A is dissipative.Uhen X is real and partially ordered by a proper uedge K,an interesting question is under uhat conditions on A the semigroup T. is positive (i.e.T fc K £ K for every fc 2 0) and also uhen T. is a positive contraction semigroup.This problem originates from the probability theory uhere positive contraction semigroups on function spaces are called Markov serai groups.Thair generators uere characterized by Feller ( £9.7,see also Oynkin i?J, 2.20 ).In C13J Phillips studied positive contraction semigroups on Banach lattices and introduced the class of the so-called dispersive operators as generators of such semigroups.They uere defined in terms of an appropriate seminorr connected uith the-positive cone.In this setting a