Generation Theorems of Semi-Groups of Linear Operators
Isao Miyadera, Shinnosuke Ôharu, Noboru Okazawa · Publications of the Research Institute for Mathematical Sciences · 1972
This paper concerns the generation of semi-groups of linear operators in a Banach space X.By a semi-group {T(t}\ t^>Q} on X we mean a one-parameter family of bounded linear operators T(t\ £^>0, such that T(0)=J (the identity operator), T(t + s)= T(t) T(s) for t, s]>0 and such that for each x G X, T(t)x is strongly continuous in £>0.For a given semi-group {T(t); t SjO} on X, we define the infinitesimal generator A Q by A 0 x = lim/j^0+ h~l(T(k)x -x} whenever the limit exists.We wish to investigate the structure and properties of {T(z); £j>0} through those of A Q .While it is desirable that A 0 has nice properties, A Q is not necessarily closed and the domain D(Ao) is not dense in X in general.In fact, a semi-group of class (0, A) is of class (0, Ci) if and only if A Q is closed, see Phillips UllU; an interesting semi-group on X with the infinitesimal generator A Q such that is discussed in Lagnese [JT].In order to investigate the properties of A Q , we consider two kinds of modified generators.One of them will be called the infinitesimal generator in the sense of Feller and the other the complete infinitesimal generator.Our first purpose is to study the basic properties of these generators.The study of the former generator is connected to the work of Feller Q3].To consider the later generator we need to make an additional assumption on the Laplace transform of the semi-group.These generators have dense domains if and