Bernoulli 泛函空间中广义计数算子的表示  

The representation of generalized number operator acting on the Bernoulli functionals space

作　　者：周玉兰 陈嘉 孔华芳 薛蕊 程秀强 ZHOU Yulan;CHEN Jia;KONG Huafang;XUE Rui;CHENG Xiuqiang(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,甘肃兰州730070

出　　处：《浙江大学学报（理学版）》2022年第3期316-323,共8页Journal of Zhejiang University（Science Edition）

基　　金：国家自然科学基金地区科学基金项目(11861057)

摘　　要：得到离散时间正规鞅平方可积泛函空间L2(M)中广义计数算子N_(h)的5种表示:(1)量子Bernoulli噪声(quantum Bernoulli noises,QBN){∂_(k),∂_(k)^(*);k≥0}的加权表示;(2)N_(h)的谱表示,广义计数算子N_(h)以h-计数测度#h的值域为其点谱;(3)N_(h)的“对角化”表示,N_(h)可表示为L^(2)(M)的标准正交基{Zσ;σ∈Γ}所生成的一维对角化正交投影算子的加权极限;(4)广义Skorohod积分-广义随机梯度表示,N_(h)可表示为互共轭算子δ_(h)和■_(h)的复合算子;(5)对N上的任意非负函数h,可构造一列有界广义计数算子,N_(h)恰为该有界广义计数算子的强极限,当h可和时,N_(h)为该有界广义计数算子的一致极限。This paper presents five representations for the generalized number operator N_(h)defined in L^(2)(M),the space of square integrable functionals in terms of the discrete-time normal martingale,(1)The weighted representation of the quantum Bernoulli noises(QBN){∂_(k),∂_(k)^(*);k≥0};(2)The spectrum representation,the spectrum of N_(h)is just the range of the h-counting measure#_(h) onΓ;(3)The"diagonalization"representation,i.e.,N_(h)can be expressed as the weighted limit of the one-dimensional diagonalized orthogonal projection operators generated by the QNB{Z_(σ);σ∈Γ};(4)The representation in terms of the generalized Skorohod integral-generalized stochastic gradient,specifically,N_(h)is the composition of the generalized Skorohodδ_(h) and its adjoint■_(h),the generalized stochastic gradient;(5)For many nonnegative function h on N,a bounded generalized number operators are constructed,which is convergent strongly to N_(h)and if h is summable,the sequence is convergent uniformly to N_(h).

关 键 词：算子谱 广义计数算子 对角化算子 广义Skorohod积分 广义随机梯度 

分 类 号：O211[理学—概率论与数理统计]