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 $\{N_{h};h\in\mathcal{P_{+}}(\mathbb{N})\}$ . $N_{h}$ 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 $\mathbb{N}$ 上非负可和函数空间 $l^{1}_{+}(\mathbb{N})$ 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 $\{N_{h};h\in\mathcal{P_{+}}(\mathbb{N})\}$ 是 $\mathcal{S}_{0}(M)$ 中的一族可交换观测.

A family of linear operators $\{N_{h};h\in\mathcal{P_{+}}(\mathbb{N})\}$ 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 $l^{1}_{+}(\mathbb{N})$ to the subspace of bounded generalized number operators on $L^{2}(M)$ , where $l^{1}_{+}(\mathbb{N})$ is the space of the summable function on $\mathbb{N}$ . We consider the conditions such that $\{N_{h_{n}};n\geqslant1\}$ is strongly and uniformly convergent. If $\{h_{n};n\geqslant1\}$ is convergent monotonically to $h$ , the domain of $\{N_{h_{n}};n\geqslant1\}$ and $N_{h}$ have some interesting properties, we show, furthermore, that a convergent family of $\{N_{h_{n}};n\geqslant1\}$ can be obtained. We prove that $\{N_{h};h\in\mathcal{P_{+}}(\mathbb{N})\}$ is commutative observable on $\mathcal{S}_{0}(M)$ .