Weak～＊ Continuous Operator Semigroups On Dual Space

A basic problem in the studying of operator semigroup theory is the relationships between semigroups and their infinitisimal generators. Given a semigroup we can define its generator. Looking back the researches on semigroup theory, no matter which type the semigroup is, a central researching problem is the generator theorem and all kinds of properties of a semigroup. No discussion on generators, the studying on semigroup theory will loss its necessary basis. This paper also starts with this point, we introduce the concepts of weak* continuous operator semigroup and its generator on the dual of a Banach space, and discuss its relevant contents.The following are the structure and main contents of this paper.In the first chapter, we discuss the dual semigroup {T*(t)}t 0 of a C0- operator semigroup {T(t)}t 0 which defined on Banach space X. In general, {T(t)}t 0 is strongly continuous, but {T*(t)}t 0 is not necessarily strongly continuous, it is weak* continuous. Meanwhile, the infinitisimal generator of {T*(t)}t 0 is the dual operator A* of A, the generator of {T(t)}t 0. And A* is weak* densely defined, weak* closed on X* . For the resolvent set and resolvent of A and A*, we have p(A) C p(A*} and R(X,A*) = R(X,A)*. A few simple properties of {T*(t)}t 0 are discussed. Finally, we prove that {T*(t)}t 0 is strongly continuous on Y* = D(A*} , and in this case the generator isThe second is the central part of the paper. We discuss the relevant properties of weak* continuous operator semigroups on X* . We introduce the concepts of weak* continuous operator semigroup and its generator. Using generator, we characterise weak* continuous operator semigroups. Firstly, a bounded linear operator A can uniquely decide a weak* continuous operator semigroup {T(t) = etA}t 0 on X* . If A is the generator of a weak* continuous operator semigroup {T(t)}t 0, then A is weak* densely defined, weak* closed; Secondly, we discuss the uniqueness of weak* continuous operator semigroups on X* . Suppose that {S(t)}t 0 and {T(t)}t 0 are weak* continuous operator semigroups on X* , several conditions are given when S(t) = T(t); Finally, we discuss the resolvent R( , A) and the resolvent set p(A) of the generator A of {T(t)}t 0, by the relevant properties of R(X,A) and p(A), we verify that a linear operator A is the infinitisimal generator of a weak* continuous operator semigroup on A'* if and only if A is weak* densely defined, weak* closed and In the last chapter, using the dissipative operators defined on X*. we characterise weak* continuous operator semigroups further. A sufficient and necessary condition about a linear weak* densely defined operator is the generator of a contraction weak' continuous operator semigroup on X* is given. Finally, for a weak* densely defined, weak* closed linear operator A on X*, if A and A* are all dissipative, then A is the generator of a contraction weak* continuous operator semigroup on X*.