离散时间正规鞅泛函空间中的广义计数算子  

Generalized number operators defined in the space of a discrete time normal martingale functional

作　　者：周玉兰 孔华芳 程秀强 薛蕊 陈嘉 ZHOU Yulan;KONG Huafang;CHENG Xiuqiang;XUE Rui;CHEN Jia(College of Mathematics and Statistics,Northwest Normal University,Lanzhou 730070,China)

机构地区：[1]西北师范大学数学与统计学院,兰州730070

出　　处：《华东师范大学学报（自然科学版）》2022年第4期13-25,共13页Journal of East China Normal University(Natural Science)

基　　金：国家自然科学基金(11861057)。

摘　　要：在离散时间正规鞅平方可积泛函空间L^(2)(M)中引入了一族线性算子{N_(h);h∈P+(N)}.N_(h)是L^(2)(M)中正的、稠定、自伴闭线性算子,一般未必有界.给出N_(h)有界的充分必要条件;讨论了N_(h)对h的依赖性,即N_(h)是关于h严格单调递增的算子值映射;证明了N上非负可和函数空间l^(1)_(+)(N)与有界广义计数算子族所成子空间是等距的;讨论了广义计数算子列强收敛和一致收敛的条件;对单调收敛函数列,讨论了其定义域收敛的条件和相应广义计数算子列收敛的条件;最后证明了{N_(h);h∈P+(N)}是S_(0)(M)中的一族可交换观测.A family of linear operators{N_(h);h∈P+(N)}in L^(2)(M)are defined.Firstly,we prove that N_(h)is a positive,densely defined,self-adjoint closed linear operator.In general,N_(h)is not bounded,hence,we explore the sufficient and necessary conditions such that N_(h)is bounded.Secondly,we consider the dependence of N_(h)on h:N_(h)is strictly increasing with respect to h,and the operator-valued mapping N_(h)is an isometry from l^(1)_(+)(N)to the subspace of bounded generalized number operators on L^(2)(M),where l^(1)_(+)(N)is the space of the summable function on N.We consider the conditions such that{N_(h_(n));n≥1}is strongly and uniformly convergent.If{h_(n);n≥1}is convergent monotonically to h,the domain of{N_(h_(n));n≥1}and N_(h)have some interesting properties,we show,furthermore,that a convergent family of{N_(h_(n));n≥1}can be obtained.We prove that{N_(h);h∈P+(N)}is commutative observable on S_(0)(M).

关 键 词：广义计数算子 算子收敛性 可交换观测 

分 类 号：O211.6[理学—概率论与数理统计]