Operator Representations On C(K)

This paper discusses the relations between some operators on C(K)and their representations in measures.At the right beginning we discuss the representationg theorems for general operators on C(K),while the transformation of of operators on C(K)to those on C(D2)are involved.Then the representation results for 1-dimensional and finite dimensional operators T are discussed,for which the related properties of its adjoints T^* and T^**are explained.In Section 4 and Section 6 the discussion is extended to the case that the set of representation measures is countable,and then is summed up as a special case thatμ_x＜＜μfor all representation measuresμ_x.In particular the properties of countable purely atomic operators are considered.By the needs of this paper,we also given the representation theorems for operators on B(K).In the last two sections(Section 7 and Section 8)the relations between compact or weakly compact operators and their representation in measures are discussed,and some characteristic properties are given.