Bernoulli泛函空间中广义计数算子的表示

得到离散时间正规鞅平方可积泛函空间L^{\mathrm{2}}(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^{\mathrm{*}};k\ge \mathrm{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^{\mathrm{2}}(M)的标准正交基\{Z_{\sigma };\sigma \in \mathrm{\Gamma }\}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{\delta }_h和{?}_h的复合算子；（5）对\mathbf{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。

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

This paper presents five representations for the generalized number operator N_h defined in L^{\mathrm{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^{\mathrm{*}};k\ge \mathrm{0}\}; (2) The spectrum representation, the spectrum of N_h is just the range of the h-counting measure {\#}_h on \mathrm{\Gamma }; (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_{\sigma };\sigma \in \mathrm{\Gamma }\}; (4) The representation in terms of the generalized Skorohod integral-generalized stochastic gradient, specifically, N_h is the composition of the generalized Skorohod {\delta }_h and its adjoint {?}_h, the generalized stochastic gradient; (5) For many nonnegative function h on \mathbf{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.