The Operators And The Integrations On Interacting Fock Space L~2(?,{?_n})

In this paper,we present the relationship of the integrations and the connec-tion between intergals and operators on interacting Fock space l~2(?,{?_n}).First,we show how to define the interacting Fock space l~2(?,{An})based on l~2(N),and in-troduce the creator operator and the annihilator operator,present the properties of its;then,we emerge the concept of the time integration?the Bochner integra-tion?the Skorohod integration?the Ito integration and the adapted gradient oper-ator?the stochastic gradient operator?the adapted projection;Finally,we study the connection between the integrations and the operators.We examine that the annihilator operator is a bounded linear unit operator,the same to the creator operator.In addition,they have the canonical commutation relations in different positions on one hand,they satisfy the canonical anti-commutation relations in the same position on the other hand.We also discover that X:? × N ? C be an adapt-ed map,Ito integral?Skorohod integral and square integral are equivalent;there is following relationship:the square integration ? the locally Bochner integration ?the Bochner integration ? the absolutely time integration.