JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE) ›› 2018, Vol. 53 ›› Issue (12): 62-68.doi: 10.6040/j.issn.1671-9352.0.2018.179

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

ZHOU Yu-lan, LI Zhuan*, LI Xiao-hui   

  1. College of Mathematics and Statistics, Northwest Normal University, Lanzhou 730070, Gansu, China
  • Online:2018-12-20 Published:2018-12-18

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+}.

Key words: Guichardet-Fock space, modified stochastic gradient, modified point state stochastic gradient, the adjoint of modified point state stochastic gradient

CLC Number: 

  • O211
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