The Operators And The Integrations On Interacting Fock Space L~2(Γ,{λ_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(Γ,{λ_n}).First,we show how to define the interacting Fock space l~2(Γ,{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:Γ × 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.