Let S and T be quasisimilar operators. By introducing the concepts of the RT-class operator, the -R-class operator and the RR-class operator, this paper gives some sufficient and necessary conditions under which each component of σre(5) intersects σre(T), and gives some sufficient conditions under which each component of σre(S) intersets some subsets of σ3(T), and also gives some sufficient and necessary conditions under which an operator is an RT-class operator and an RR-class operator respectively.