Abstract:

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 $\mathcal{S}_{0}(M)$ .

Key words: generalized number operator, convergence of operators, commutative observables