In Bratelli Robinson the number operator in Fock space is defined as: $$\mathcal{D}(N):=\{\phi\in\mathcal{F}:\sum_{n=1}^\infty n|\|\phi_n\|<\infty\}\\ N:\mathcal{D}(N)\to \mathcal{F}:\phi\mapsto\sum_{n=1}^\infty n\phi_n$$ Is this operator closed?
Moreover, if I would define instead: $$\mathcal{D}(N_0):=\{\phi\in\mathcal{F}:\phi_n=0\text{ a.a}\}\\ N_0:\mathcal{D}(N_0)\to \mathcal{F}:\phi\mapsto\sum_{n=1}^N n\phi_n$$ Would this one be closable? (Here I'm not sure wether $\phi_\lambda\to 0$ and $N_0\phi_\lambda\to\psi$ imply $\psi=0$.)
"Yes, the number operator is closed!"
The number operator is self adjoint, namely if $\psi\in\mathcal{F}$ is such that there exists $\eta\in\mathcal{F}$ with $\langle\psi,N\phi\rangle=\langle\eta,\phi\rangle$ for all $\phi\in\mathcal{D}(N)$, then $\psi\in\mathcal{D}(N)$ and $\eta=N\psi$. Especially the number operator is closed.
"Yes, the restriction is closable."
The domain is a core for the number operator, namely if $\psi\in\mathcal{D}(N)$ is such that $\langle\psi,\phi\rangle+\langle N\psi,N_0\phi\rangle=0$ for all $\phi\in\mathcal{D}(N_0)$, then $\psi=0$ *. Especially the restriction is closable.