连续时间Guichardet-Fock空间中的计数算子的表示 |
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引用本文: | 周玉兰,李晓慧,程秀强,薛蕊. 连续时间Guichardet-Fock空间中的计数算子的表示[J]. 山东大学学报(理学版), 2019, 54(11): 108-114. DOI: 10.6040/j.issn.1671-9352.0.2019.513 |
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作者姓名: | 周玉兰 李晓慧 程秀强 薛蕊 |
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作者单位: | 西北师范大学数学与统计学院, 甘肃 兰州 730070 |
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基金项目: | 国家自然科学基金地区科学基金资助项目(11461061) |
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摘 要: | 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=δSymbolQC@;其次,应用L2(Γ;η)中有界算子族{SymbolQC@*sSymbolQC@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@*sSymbolQC@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=1nJn,其中Jn:L2(Γ;η)→L2(Γ(n);η)是正交投影。
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关 键 词: | 修正随机梯度SymbolQC@ 修正点态随机梯度SymbolQC@s 修正点态随机梯度的共轭SymbolQC@*s Skorohod积分δ 计数算子N |
Representation of the number operator in continuous-time Guichardet-Fock space |
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Affiliation: | College of Mathematics and Statistics, Northwest Normal University, Lanzhou 730070, Gansu, China |
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Abstract: | 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=δSymbolQC@. Secondly, the representation of Bochner integral is given: N=∫R+SymbolQC@*sSymbolQC@sds in the sense of the inner product, by means of the family of isometric operator {SymbolQC@*sSymbolQC@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=1nJn, where Jn:L2(Γ;η)→L2(Γ(n);η), is the orthogonal projection. |
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Keywords: | modified stochastic gradient SymbolQC@ point-state modified stochastic gradient SymbolQC@s adjoint of the point state modified stochastic gradient SymbolQC@*s Skorohod integral δ number operator N |
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