On The Study Of Several Kinds Of Cosine Operator Functions Topology

In this paper, on the basis of the theory of locally convex linear topological spacesand the theory of semigroups of bounded linear operators, combined with several kindsof cosine operator functions, cosine operator functions topology, weak cosine operatorfunctions topology, C-cosine operator functions topology, weak C-cosine operatorfunctions topology, integrated cosine operator functions topology, weak integrated cosineoperator functions topology, integrated C-cosine operator functions topology, weakintegrated C-cosine operator functions topology are put forword, and basic properties ofthem are given. So the theory of semigroups topology is enriched. This paper mainresearch content to be as follows:1．Quasi-norms are induced by using cosine operator functions and continuouslinear functional, the concept of cosine operator functions topology and weak cosineoperator functions topology are given, and research their basic properties:completeness,separability, strong and weak of topology.2．Quasi-norms are induced by using C-cosine operator functions and continuouslinear functional, the concept of C-cosine operator functions topology and weakC-cosine operator functions topology are given, and research their basic properties:completeness, separability, strong and weak of topology.3．Quasi-norms are induced by using integrated cosine operator functions andcontinuous linear functional, the concept of integrated cosine operator functions topologyand weak integrated cosine operator functions topology are given,and research their basicproperties:completeness, separability, strong and weak of topology.4．Quasi-norms are induced by using integrated C-cosine operator functions and continuous linear functional, the concept of integrated C-cosine operator functionstopology and weak integrated C-cosine operator functions topology are given,andresearch their basic properties:completeness, separability,strong and weak of topology.5．Among the study of the four parts, the relationships of several kinds of cosineoperator functions topology are studied.