连续时间Guichardet-Fock空间中修正随机梯度算子的性质
Properties of modified stochastic gradient operators in continuous-time Guichardet-Fock space

摘要： 讨论了连续时间Guichardet-Fock空间L2(Γ;η)中修正随机梯度算子及修正点态随机梯度算子族{s;s∈R+}的性质。讨论表明:修正随机梯度算子是L2(Γ;η)中的稠定无界线性算子,而修正点态随机梯度算子族{s;s∈R+}及其共轭族{*s;s∈R+}是L2(Γ;η)中的有界线性算子,具有很多性质:满足典则反交换关系和幂零性;{s;s∈R+}与{*s;s∈R+}的不等时复合可交换,即s*s=*ss,对∠s≠t;同时{*ss;s∈R+}是L2(Γ;η)上一族正交投影。另外,利用{s;s∈R+}和{*s;s∈R+},构造了L2(Γ;η)上一个酉算子群。
Abstract: The paper investigate the properties of the modified stochastic gradient operator and modified point-state stochastic gradient operators {s;s∈R+} in continuous-time Guichardet-Fock space L2(Γ;η). We show that the modified stochastic gradient operator is a unbounded, densely defined linear operator in L2(Γ;η); the family of modified point-state stochastic gradient operators {s;s∈R+} and its adjoint {*s;s∈R+} are bounded linear operator, which have many properties. For example, they satisfies the canonical anti-commutation relations(CAR)and nilpotency; s*s=*ss, for ∠s≠t, which means that, the family of operators{s;s∈R+} and {*s;s∈R+} are commutive for ∠s≠t; the operator {*ss;s∈R+} is a family of orthogonal projections on L2(Γ;η). Meanwhile, we construct a unitary operator group on L2(Γ;η) with the point-state modified stochastic gradient {s;s∈R+} and its adjoint {*s;s∈R+}