On The Study Of Several Semigroups Topology

The topology of locally convex linear space can be established by a family of seminorms,but different seminorms can be induced by using operator semigroup , and then the different locally convex linear topological space can be established,moreover they have some special properties.It has been studied by professor Zhao Huaxin in 2006, and the theories of semigroup topology are introduced.In this paper,based on the theories of semigroup topology,combines with several semigroupsof linear operators , generalized C₀-semigroup , integrated semigroup , T(t)-integratedsemigroup , integrated C-semigroup, generalized C₀-semigroup topology , integratedsemigroup topology , T(t)-integrated semigroup topology, integrated C-semigroup topology are put forword,their basic properties as weel as the properties of the new topologicalmeaning are given.This article main research content and makes the progress to be as follows:1.Seminorm p_t(x)=‖CT(t)x‖,x∈X,t≥0 be induced by using generalized C₀-semigroup {T(t),t≥0} the concept of generalized C₀-semigroup topology are given , and researched its basic properties:base of seminorm, completeness,separability,stong topology as weel as the properties of the new topological meaning are given.2.Seminormp_λ(x)=‖R(λ,A)x‖=‖λⁿ integral from n=0 to∞e^-λtS(t)xdt‖,x∈X,t≥0 be inducedby using integrated semigroup{T(t),t≥0},the concept of integrated semigroup topology are given,and research its properties:completeness,separability,stong topology .3.Seminorm p_λ(x)=‖R(λ,A)L_λx‖=‖λⁿ integral from n=0 to∞e^-λt S(t)xdt, x∈X,t≥0 beinduced by usingT(t)-integrated semigroup{T(t),t≥0},the concept of T(t)-integrated semigroup topology are given,and researched its basic properties:completeness,separability, stong topology .4.Seminorm p_λ(x)=‖R(λ,A)C_x‖=‖λⁿ integral from n=0 to∞e^-λt T(t)xdt, x∈X,t≥0 be inducedby using integrated C-semigroup{T(t),t≥0},the concept of integrated C-semigroup topology are given,and research its properties:completeness,separability,stong topology .5.Among the researched of the five pates,the relationships topology and stong topologyof opertors are studied.