Multi-Parametes C Semigroups Relevant Issues

On the basis of the definition single-and-double parameters C semigroups and its infinitesimal generator,generating theorem,Yosida approximation,this paper uses the methods of studying classical operator semigroups.According to the previous study of single-and-double parameters C semigroups,this paper discussed the definition of multi-parameters C semigroups and the basic properties of its generator,and studies its infinitesimal generator,generating theorem,and the Hille-Yosida theorem and approximation,as well as its expression(index formula)and spectrum.The article is divided into several aspects:1.The definition of multi-parameters C semigroups and the representation of infinitesimal generator of multi-parameters C semigroups are given.The properties of multi parameters C semigroups infinitesimal generator and its related conclusion are discussed.2.According to the proof of generating theorem for single-and-double parameters C semigroups their generating theorem are extended to multi-parameters C semigroups,and the Hille-Yosida theorem of multi-parameters C semigroups is also proved.3.The Hille-Yosida approximation of multi-parameters C semigroups is discussed and proved,and the conditions for differentiability of multi-parameters C semigroups are given.4.Based on the properties of single-and-double parameters C semigroups and the form of exponential formula,the expression of multi-parameters C semigroups is given.5.The spectrum of the multi-parameters C semigroups is studies.The concept of the spectrum of multi-parameters C semigroups(continuous spectrum,point spectrum,residual spectrum)is given,and the related properties and theorem of multi-parameters C semigroups are deduced.