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

Continuous-time Guichardet-Fock space L2(?;?)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 ?:N=???.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(?(n);?)of L2(?;?)is just the eigenspace corresponding to the eigenvalue n,then N has the spectrum representation:(?),where Jn:L2(?;?)?L2(?(n);?)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(?;?).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(?;?)?L2(?t(n);?)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=?(I[0,t)??=???I[0,t].