离散时间正规鞅泛函空间中幂计数算子的性质  

Properties of power-number operators in the functional space of discrete time normal martingale

作　　者：周玉兰 魏万瑛 柳翠翠 杨青青 ZHOU Yulan;WEI Wanying;LIU Cuicui;YANG Qingqing(College of Mathematics and Statistics,Northwest Normal University,Lanzhou 730070,Gansu,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：《山东大学学报(理学版)》2025年第2期85-95,共11页Journal of Shandong University(Natural Science)

基　　金：国家自然科学基金资助项目(12261080)。

摘　　要：在正规鞅平方可积泛函空间L^(2)(M)中引入一类稠定自伴线性算子a^(N),其中a是任意正实数,N是L^(2)(M)中计数算子,称a^(N)为N的a级幂计数算子。首先讨论a^(N)的分析性质:给出a^(N)有界的充要条件,且发现当a^(N)有界时都是单位算子。其次给出a^(N)是紧算子的充要条件;讨论{a^(N);a>0}的算子结构及其谱分析:a^(N)的谱集是{a^(n);n>0},其特征向量全体恰好构成L^(2)(M)的一组标准正交基,且1是它们的公共谱点,对应1的唯一特征向量是L^(2)(M)的真空态Z_(∅)。然后讨论a^(N)对a的依赖性。最后应用Γ-指标集量子Bernoulli噪声,对任意a∈(0,1),构造了a^(N)的一致收敛序列,而当a>1时,构造了a^(N)的强收敛序列。A class of densely defined self-adjoint linear operator a^(N) is introduced in the functional space L^(2)(M)of normal martingale square-integrable,where a is a positive number,and N is the number operator in L^(2)(M),a^(N) is called the a-level power-number operator of N.Firstly,the analytical properties of a^(N) are discussed:a sufficient and necessary condition that a^(N) is bounded is given,and in this case,{a^(N);a>1}are unit operator on L^(2)(M).Secondly,a^(N) is compact operator if and only if O<a<l;the construction and the spectrum of {a^(N);a>0}are discussed:{a^(n);a>0}is the spectrum of a^(N) and all of its eigenvector forms an orthonormal basis of L^(2)(M),1 is the unique spectrum of{a^(N);a>0} and the vacuum Z_(∅) is the unique common eigenvector of 1.And then the dependence of a^(N) on a is discussed.Finally,a uniform convergence sequence of a^(N) for a ∈(O,1)and a strong convergence sequence of a^(N) is constructed when a>1 by means of the quantum Bernoulli noise indexed by Γ.

关 键 词：幂计数算子 算子谱 紧算子 Γ-指标集量子Bernoulli噪声 算子收敛 

分 类 号：O177[理学—数学] O211.63[理学—基础数学]