Research On Related Problems Of N-th Order α-Times Integral C-Semigroup

The relationship between operator semigroups and their sub-generators is an important part of the theory of operator semigroups.The study of n-th order α-times integral C-semigroup and its sub-generator、perturbations、spectrum is helpful for the subsequent study of the related theory of the n-th order α-times integral C-semigroup.In this thesis,the generation theorem of bi-continuous n-th order α-times integral C-semigroup is discussed first;then the perturbation of one-parameter n-th order α-times integral C-semigroup and the perturbation of two-parameter n-th orderα-times integral C-semigroup are studied;finally,the spectral mapping theorem of one-parameter n-th order α-times integral C-semigroup and the spectral mapping theorem of two-parameter n-th order α-times integral C-semigroup are proven.This thesis consists of the following three parts:In the first part,the resolvent and the resolvent set of bi-continuous n-th order α-times integral C-semigroup are defined,and the generation theorem of bi-continuous n-th order α-times integral C-semigroup is discussed.In the second part,perturbations of one-parameter n-th order α-times integral C-semigroup and the two-parameter n-th order α-times integral C-semigroup are discussed,which generates two important results.In the third part,the concepts of the point spectrum,the continuous spectrum and the residual spectrum of one-parameter n-th order α-times integral C-semigroup are given.The spectral mapping theorem of one-parameter n-th order α-times integral C-semigroup and the spectral mapping theorem of two-parameter n-th order α-times integral C-semigroup are discussed.