Fock sector projections

[Algebraic properties] For $\psi \in F(H)$ the tuple ${\mathrm{\Pi }}_n\psi$ lies in $F(H)$ because its coordinate $n$ is ${\psi }_n\in \mathrm{ran}{P}_{+}^{(n)}$ and all other coordinates vanish ([F1]); clearly ${\mathrm{\Pi }}_n(\psi +c\chi )={\mathrm{\Pi }}_n\psi +c{\mathrm{\Pi }}_n\chi$, so ${\mathrm{\Pi }}_n$ is linear, and $\mathrm{\parallel }{\mathrm{\Pi }}_n\psi {\mathrm{\parallel }}^2=\mathrm{\parallel }{\psi }_n{\mathrm{\parallel }}^2\le \mathrm{\parallel }\psi {\mathrm{\parallel }}^2$ gives $\mathrm{\parallel }{\mathrm{\Pi }}_n\mathrm{\parallel }\le 1$. Obviously ${\mathrm{\Pi }}_n^2={\mathrm{\Pi }}_n$. For the adjoint: $⟨{\mathrm{\Pi }}_n\psi ,\phi ⟩=⟨{\psi }_n,{\phi }_n⟩=⟨\psi ,{\mathrm{\Pi }}_n\phi ⟩$ for all $\psi ,\phi \in F(H)$, so ${\mathrm{\Pi }}_n^{\ast }={\mathrm{\Pi }}_n$ by uniqueness of the Hilbert-space adjoint.

[Orthogonality and ranges] For $n\mathrm{\ne }m$, ${\mathrm{\Pi }}_n{\mathrm{\Pi }}_m=0$ because the tuple ${\mathrm{\Pi }}_m\psi$ is supported in coordinate $m$ and ${\mathrm{\Pi }}_n$ reads coordinate $n$. The range of ${\mathrm{\Pi }}_n$ is exactly the set of vectors of $F(H)$ supported in coordinate $n$, a closed subspace that is identified with $\mathrm{ran}{P}_{+}^{(n)}$ by the isometry ${\iota }_n$ ; it is closed in $F(H)$ because if ${\mathrm{\Pi }}_n{\psi }^{(k)}\to \eta$, then ${\eta }_m=0$ for $m\mathrm{\ne }n$ and ${\eta }_n\in \mathrm{ran}{P}_{+}^{(n)}$ by closedness ([F2]). For every $\psi$, the difference $\psi -{\mathrm{\Pi }}_n\psi$ has zero coordinate $n$ and is orthogonal to every vector supported there by the direct-sum inner product; hence [F3] identifies ${\mathrm{\Pi }}_n$ with the orthogonal projection onto its range. This range identification also proves ${\mathrm{\Pi }}_n=0$ exactly when the sector is zero; when $\mathrm{ran}{P}_{+}^{(n)}\mathrm{\ne }\{0\}$ choose a unit $u$ there and note $\mathrm{\parallel }{\mathrm{\Pi }}_n{\iota }_n(u)\mathrm{\parallel }=1$, so $\mathrm{\parallel }{\mathrm{\Pi }}_n\mathrm{\parallel }=1$. For $H\mathrm{\ne }\{0\}$ and $n\ge 1$ one has $0\mathrm{\ne }{e}^{\otimes n}\in \mathrm{ran}{P}_{+}^{(n)}$ for any $e\mathrm{\ne }0$ in $H$, and $\mathrm{ran}{P}_{+}^{(0)}=\mathbb{C}$, so no ${\mathrm{\Pi }}_n$ vanishes. If $H=\{0\}$, every positive tensor power is zero, $F(H)$ consists only of its scalar vacuum coordinate, ${\mathrm{\Pi }}_0=I$, and ${\mathrm{\Pi }}_n=0$ for $n\ge 1$; the same coordinate arguments apply. Thus (b), (d) hold.

[Orthogonal expansion] For $\psi \in F(H)$ put ${\psi }^{(N)}:={\sum }_{n\le N}{\mathrm{\Pi }}_n\psi$ and $r^{(N)}:=\psi -{\psi }^{(N)}$, whose coordinates are ${\psi }_n$ for $n>N$ and $0$ otherwise. Then $\mathrm{\parallel }{r}^{(N)}{\mathrm{\parallel }}^2={\sum }_{n>N}\mathrm{\parallel }{\psi }_n{\mathrm{\parallel }}^2\to 0$ by [F1], so ${\sum }_n{\mathrm{\Pi }}_n\psi$ converges to $\psi$ in norm. The terms ${\mathrm{\Pi }}_n\psi ={\iota }_n({\psi }_n)$ are pairwise orthogonal by step 1.2, and [F4] gives $\mathrm{\parallel }{\psi }^{(N)}{\mathrm{\parallel }}^2={\sum }_{n\le N}\mathrm{\parallel }{\psi }_n{\mathrm{\parallel }}^2$; passing to the limit and using continuity of the norm and of the inner product gives $\mathrm{\parallel }\psi {\mathrm{\parallel }}^2={\sum }_n\mathrm{\parallel }{\mathrm{\Pi }}_n\psi {\mathrm{\parallel }}^2$.

[Strong convergence and conclusion] Since $\parallel \psi -{\sum }_{n\le N}{\mathrm{\Pi }}_n\psi {\parallel }^2={\sum }_{n>N}\mathrm{\parallel }{\psi }_n{\mathrm{\parallel }}^2\to 0$ for every $\psi \in F(H)$ by step 2.1, the partial sums of $({\mathrm{\Pi }}_n)$ converge to the identity in the strong operator topology; combined with steps 1.1, 1.2 and 2.1 this proves (a)-(d).