连续时间Guichardet-Fock空间中的计数算子的表示

考虑了连续时间Guichardet-Fock空间L2(Γ;η)中计数算子N的表示问题。利用修正随机梯度SymbolQC@及非适应性Skorohod积分δ,给出N的梯度-积分表示:N=δSymbolQC@;其次,应用L²(Γ;η)中有界算子族{SymbolQC@^*_sSymbolQC@_s;s∈R₊}的算子积分,证明在弱意义下,N有有界算子族的Bocher积分表示:N=∫_R+SymbolQC@^*_sSymbolQC@_sds;同时,发现L²(Γ;η)的一列相互正交闭子空间L²(Γ⁽ⁿ⁾;η)是N的特征子空间,从而给出N的谱表示:N=∑^∞_n=1nJ_n,其中J_n:L²(Γ;η)→L²(Γ⁽ⁿ⁾;η)是正交投影。

Representation of the number operator in continuous-time Guichardet-Fock space

The paper considers the representation of the number operator N in continuous-time Guichardet-Fock space L2(Γ;η). Firstly, the gradient-Skorohod integral representation of N is given by using modified stochastic gradient SymbolQC@ and non-adaptive Skorohod integral δ:N=δSymbolQC@. Secondly, the representation of Bochner integral is given: N=∫_R+SymbolQC@^*sSymbolQC@sds in the sense of the inner product, by means of the family of isometric operator {SymbolQC@^*sSymbolQC@_s; s∈R₊}. Meanwhile, the spectrum of N is just the nonnegative integral N, and for any n≥0, the closed subspace L2(Γ⁽ⁿ⁾;η) of Guichardet-Fock space L2(Γ;η) is just the eigenspace corresponding to the eigenvalue n, and N has the spectrum representation: N=∑^∞_n=1nJ_n, where J_n:L2(Γ;η)→L2(Γ⁽ⁿ⁾;η), is the orthogonal projection.