| Continuous-time Guichardet-Fock space L2(r;η)is a space consisted of square integrable function space defined Γ valued on the complex separable Hilbert spaceη,where Γ is the set of finite power on R+.The operator defined on L2(Γ,η)have maximal domains,and broken through the set of exponential vector.In this paper,we mainly investigate the properties of the generalized modified stochastic gradient▽h and the generalized Skorohod integral operator δh,and the relations between ▽h,δh and the other defined in the continuous-time Guichardet-Fock space L2(Γ,η).Firstly,we define a family of operators ▽h and δh in L2(Γ;η)and L2(Γ×R+;η),for a nonnegative function h defined on R+,which we called the generalized modified stochastic gradient ▽h and the generalized Skorohod integral operator δh.Moreover,we find that ▽h and δh are densely defined and closed in L2(Γ;η)and L2(Γ×R+;η),respectively,but generally speaking,they are unbounded,for a special kind of nonnegative function Vh and 8h are bounded,furthermore,we give an upper estimation for the norm of ▽h and δh.Secondly,we consider the representation for the generalized modified stochastic gradient ▽h and the generalized Skorohod integral operator δh in terms of the point-state modified stochastic gradient {▽s;s∈R+} and its adjoint operator {V▽s*;s∈ R+} by means of the weighted-Bochner integral of the bounded linear operator.By using the representation of the weighted-Bochner integral in terms of the point-state modified stochastic gradient {▽s;s ∈R+} and its adjoint operator {▽s*;s ∈ R+},we show that ▽h and δh are adjoint operator for each other.Finally,we discuss the relations between generalized number operator Nh and ▽h,δh. |
