连续时间Guichardet-Fock空间中修正随机梯度算子的性质

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。

Properties of modified stochastic gradient operators in continuous-time Guichardet-Fock space

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₊}.