The Repesentation Of The Number Operator And The Related Problems In The Continuous-time Guichardet-Fock Space

Continuous-time Guichardet-Fock space L2(Γ;η)is a Hilbert space with rich spatial structure,which is an important basic framework for studying spatial struc-ture and correlation operators in quantum stochastic calculus.The number operator has important research significance in many research fields such as quantum stochas-tic calculus,martingale and solution of linear stochastic Schrodinger equations.The paper considers the representation of the number operator N and the related prob-lems in continuous-time Guichardet-Fock space.Firstly,we prove that N is a densely defined,unbounded operator and give the gradient-Skorohod integral representation of N by using modified stochastic gradient ▽ and non-adaptive Skorohod integral δ:N=δ○▽.The representation of Bochner integral is given:N=∫R+▽s*▽sds in inner product,by means of the family of isometric operator {▽s*▽s;s∈R+}.Meanwhile,the spectrum of N is just the nonnegative integral N.For any n≥0,the closed subspace L2(F(Γ(n);η)of L2(Γ;η)is just the eigenspace corresponding to the eigenvalue n,then N has the spectrum representation:(?),where Jn:L2(Γ;η)→L2(Γ(n);η)is the orthogonal projection.Secondly,we consider the properties of the "number operator-valued process"{Nt}t≥0 in continuous-time Guichardet-Fock space L2(Γ;η).we prove that {Nt}t>0 is a family of densely defined,unbounded operators and N0 is an bounded and orthogonal projection.Finally,We establish three different representations of {Nt}t>0.The first one is the spectrum representation:Nt=∑n>0nJt(n),where Jt(n);L2(Γ;η)→L2(Γt(n);η)is the orthogonal projection.The second one is the Bochner integral representation Nt=f0t▽s*▽sds in inner product,by means of the family of isometric operator{▽s*▽s;s∈R+}.The third one is the gradient-Skorohod integral representation Nt=δ(I[0,t)○▽=δ○▽I[0,t].