Rays, unit vectors and rank-one Hilbert projections

[(a) trace, positivity and scaling] By [F1], $T_{\psi ,\psi }$ is trace class with $\mathrm{\parallel }{T}_{\psi ,\psi }{\mathrm{\parallel }}_1=\mathrm{\parallel }\psi {\mathrm{\parallel }}^2$ and $\mathrm{tr}(T_{\psi ,\psi })=⟨\psi ,\psi ⟩=\mathrm{\parallel }\psi {\mathrm{\parallel }}^2$, and it satisfies $T_{\psi ,\psi }^{\ast }=T_{\psi ,\psi }$. For every $x\in H$, $⟨T_{\psi ,\psi }x,x⟩=⟨⟨x,\psi ⟩\psi ,x⟩=\mathrm{\mid }⟨x,\psi ⟩{\mathrm{\mid }}^2\ge 0$ by [F6], so $T_{\psi ,\psi }$ is positive under the general bounded-operator definition in [F6]; also $T_{\psi ,\psi }\psi =\mathrm{\parallel }\psi {\mathrm{\parallel }}^2\psi \mathrm{\ne }0$, so $T_{\psi ,\psi }$ is nonzero. For $z\mathrm{\ne }0$ the conjugate-linearity of [F1] gives $T_{z\psi ,z\psi }=\mathrm{\mid }z{\mathrm{\mid }}^2{T}_{\psi ,\psi }$; substituting $z=\mathrm{\parallel }\psi {\mathrm{\parallel }}^{-1}$ and using $\nu (\psi )=\mathrm{\parallel }\psi {\mathrm{\parallel }}^{-1}\psi$ gives $T_{\nu (\psi ),\nu (\psi )}=\mathrm{\parallel }\psi {\mathrm{\parallel }}^{-2}{T}_{\psi ,\psi }$, that is the displayed identity after multiplying by $\mathrm{\parallel }\psi {\mathrm{\parallel }}^2$.

[(b) $T_{u,u}$ is the projection onto $\mathbb{C}u$] Let $u\in S(H)$. The line $\mathbb{C}u$ is a closed linear subspace by [F5]. For every $x$ one has $T_{u,u}x=⟨x,u⟩u\in \mathbb{C}u$, and $⟨x-T_{u,u}x,u⟩=⟨x,u⟩-⟨x,u⟩\mathrm{\parallel }u{\mathrm{\parallel }}^2=0$ by [F6], so $x-T_{u,u}x\in (\mathbb{C}u{)}^{\perp }$; by the characterisation of [F4] applied to $M=\mathbb{C}u$, this says $T_{u,u}=P_{\mathbb{C}u}$, so $T_{u,u}$ is a Hilbert orthogonal projection and in particular is nonzero with $T_{u,u}^2=T_{u,u}$ and $T_{u,u}^{\ast }=T_{u,u}$.

[(c) equality of rank-one operators] Suppose first that $T_{\psi ,\psi }=T_{\phi ,\phi }$ with $\psi ,\phi \mathrm{\ne }0$. Evaluating both sides at $\psi$ gives $\mathrm{\parallel }\psi {\mathrm{\parallel }}^2\psi =⟨\psi ,\psi ⟩\psi =T_{\psi ,\psi }\psi =T_{\phi ,\phi }\psi =⟨\psi ,\phi ⟩\phi$ by [F1] and [F6]; the left side is nonzero, so $⟨\psi ,\phi ⟩\mathrm{\ne }0$ and $\phi =z\psi$ with $z=\mathrm{\parallel }\psi {\mathrm{\parallel }}^2\mathrm{/}⟨\psi ,\phi ⟩\mathrm{\ne }0$. Hence $\phi$ and $\psi$ lie in the same ray, so $\mathrm{\parallel }\phi \mathrm{\parallel }=\mathrm{\mid }z\mathrm{\mid }\mathrm{\parallel }\psi \mathrm{\parallel }$ by [F6]; moreover $T_{\phi ,\phi }=T_{z\psi ,z\psi }=\mathrm{\mid }z{\mathrm{\mid }}^2{T}_{\psi ,\psi }=T_{\psi ,\psi }$ by conjugate-linearity and the hypothesis, and since $T_{\psi ,\psi }\mathrm{\ne }0$ this forces $\mathrm{\mid }z{\mathrm{\mid }}^2=1$, hence $\mathrm{\parallel }\phi \mathrm{\parallel }=\mathrm{\parallel }\psi \mathrm{\parallel }$. Conversely, if $\mathrm{\parallel }\psi \mathrm{\parallel }=\mathrm{\parallel }\phi \mathrm{\parallel }$ and $[\psi ]=[\phi ]$, then $\phi =z\psi$ for some $z\mathrm{\ne }0$, and taking norms gives $\mathrm{\mid }z\mathrm{\mid }=1$; then $T_{\phi ,\phi }=T_{z\psi ,z\psi }=\mathrm{\mid }z{\mathrm{\mid }}^2{T}_{\psi ,\psi }=T_{\psi ,\psi }$. For unit vectors this is the stated equivalence $T_{u,u}=T_{w,w}⟺w=\zeta u$, $\zeta \in U(1)$, by [F3].

[(b) the converse identification] Let $M$ be a one-dimensional linear subspace and let $u\in S(H)$ with $M=\mathbb{C}u$. For arbitrary $x$, the defining properties of $P_M$ in [F4] give $P_{M}x\in M=\mathbb{C}u$, so $P_{M}x=\alpha u$ for a unique scalar $\alpha$, and $x-P_{M}x\in M^{\perp }$, so $⟨x-P_{M}x,u⟩=0$. Consequently $\alpha =\alpha \mathrm{\parallel }u{\mathrm{\parallel }}^2=⟨\alpha u,u⟩=⟨P_{M}x,u⟩=⟨x,u⟩$, and therefore $P_{M}x=⟨x,u⟩u=T_{u,u}x$ for every $x$. Thus the Hilbert orthogonal projections with one-dimensional range are exactly the operators $T_{u,u}$, $u\in S(H)$.

[(d) the ray-to-state map] If $[\psi ]=[\phi ]$, then by [F3] the unit representatives satisfy $\nu (\phi )=\zeta \nu (\psi )$ for some $\zeta \in U(1)$, whence $T_{\nu (\phi ),\nu (\phi )}=T_{\zeta \nu (\psi ),\zeta \nu (\psi )}=\mathrm{\mid }\zeta {\mathrm{\mid }}^2{T}_{\nu (\psi ),\nu (\psi )}=T_{\nu (\psi ),\nu (\psi )}$ by the conjugate-linearity of [F1]; thus $j$ is well defined. If $T_{\nu (\psi ),\nu (\psi )}=T_{\nu (\phi ),\nu (\phi )}$, then step 2.2 applied to the unit vectors $\nu (\psi ),\nu (\phi )$ gives $\nu (\phi )=\zeta \nu (\psi )$ for some $\zeta \in U(1)$, so $[\phi ]=[\psi ]$ by [F3]; thus $j$ is injective. Its image is $\{T_{u,u}:u\in S(H)\}$ by definition of $\nu$, and by steps 2.1 and 3.1 this set consists exactly of the one-dimensional Hilbert orthogonal projections of $H$. This completes (a)-(d).