Regular Predissociation And Operator Families Spectrum Continuity

In this thesis, some properties of C-regularized resolvent family have been studied. Thiswork bases on the studying of resolvent family, it extents the concept of resolvent family of op-erator, all the conclusions in the thesis have includes the corresponding results of C₀-semigroup,C-semigroup and resolvent family . There're four chapters in this thesis.Chapter 1 is preface, it simply introduces the history and development of C-regularizedresolvent family.In Chapter 2, we first introduce some useful properties of C-regularized resolvent family .Then two definitions of infinitesimal generator are given in Section 3. Based on the spectrum ofC-regularized resolvent family, we study the relationship between spectrum of C-regularizedresolvent family and of its generator, and gain the spectral mapping theorem in section 4. In thelast section , adjoint operator of C-regularized resolvent family is studied.Chapter 3 is about continuity and stability of C-regularized resolvent family. We discussthe uniform continuity of C-regularized resolvent family on the different condition : if R(t) isexponential bounded C-regularized resolvent family on Banach space X, it's uniform continuityif and only if its generator is a bounded operator on Banach space X; if R(t) is exponentialstability on Hilbert space H, uniform continuity is equivalent to the decay to zero of a holomorphicoperator family along some imaginary axis. In last section, we give sufficient condition for theuniform stability of C-regularized resolvent family.In Chapter 4 ,all work discussed above is concluded and further study directions are putforward.