Research On Several Problems Of C-regular Presolving Operator Families

In this paper,we study the related problems of the C-regularized families,including the invariant subspace of the C-regularized families,the admissible subspace,the Hille-Yosida space and the invariant flow.The full text is divided into five chapters.In Chapter 1,we introduce the historical background of the C-regularized families and the existing invariant subspace,the allowable subspace,the Hille-Yosida space and the invariant flow of the operator semigroup.In Chapter 2,we give the basic conclusions and properties of the C-regularized families,including the two generation theorems of the C-regularized families,the exponential representation theorem of the C-regularized families,the predistortion equation of the C-regularized families,and a convergence theorem of the C-regularized families:?H(?)x?Cx.In Chapter 3,we generalize the invariant subspace and allowable subspace of C0 semigroups to the C-regularized families.First,we give the concept of the invariant subspace and the admissible subspace of the C-regularized families.Then we prove that there is a sufficient and necessary condition for the invariant subspaces of C-regular precursors.Next we have obtained two equivalent descriptions of Y being A-admissible in the general Banach space and the reflexive Banach space.Finally,we get the description of Y being A-admissible in C-regularized families in the sense of isomorphism.In Chapter 4,we discuss the Hille-Yosida space problem of the C-regularized families.First,we give the concept of the Hille-Yosida space Zk of the C-regularized families.Then we construct a set of generators Ak,which is the infinitesimal generator of the C-regular predistortion operator family of contractions on Zk,and,in a certain sense,the spaces Zk are maximal-unique.Finally,we obtain that Ak is the infinitesimal generator of the C-regular semigroups of contractions when a is positive.Chapter 5 deals with the invariant flow for nonautonomous C-regular semigroups.First of all,we give the concept of tangency,and then we illustrate the equivalence property of the invariant flow and tangency.Finally,give the proof of the equivalence.