I am trying to show that for $p \in \{0,1,2...\}$ the categories of p-multigraded and (p+2)-multigraded Hilbert spaces are equivalent. A p-multigraded Hilbert space is a graded Hilbert space $H = H^+ \oplus H^-$ equipped with $p$ odd unitary operators $\{e_i\}_{i=1}^p$ satisfying $e_i e_j = - e_je_i$ for $i \neq j$ and $e_i^2 = -1$. So we can consider $H$ as a graded left module over the complex Clifford algebra $\mathbb{C}_n$. If we let $\mathcal{H}^p$ denote the category of p-multigraded Hilbert spaces where the morphisms are bounded operators that are $\mathbb{C}_n$ linear, i.e. commute with the grading operators; $$Hom_{\mathcal{H}^p}(H,H') = \{T \in B(H,H') | Te_i = e_i'T:i=1,..,p \} $$ For p-multigraded Hilbert spaces H and H' with their respective grading operators. As far as i understand, i have to construct a functor $F: \mathcal{H}^{p+2} \to \mathcal{H}^{p}$ which satisfies
1) F is full; for $H,H' \in \mathcal{H}^{p+2}$ the map $Hom_{\mathcal{H}^{p+2}}(H,H') \to Hom_{\mathcal{H}^p}(FH,FH')$ is surjective.
2) F is faithful; for $H,H' \in \mathcal{H}^{p+2}$ the map $Hom_{\mathcal{H}^{p+2}}(H,H') \to Hom_{\mathcal{H}^p}(FH,FH')$ is injective.
3) F is essentially surjective; For any $H' \in \mathcal{H}^{p}$ there exists $H \in \mathcal{H}^{p+2}$ such that $FH \simeq H'$
I know that the F is constructed as follows: If $H$ is a (p+2)-multigraded Hilbert space with operators $\{e_i\}_{i=1}^{p+2}$, we define $e = ie_{p+1}e_{p+2}$. Since this is selfadjoint and has square 1, it has eigenvalues $\pm 1$ and H decomposes as a direct sum of the $\pm 1$-eigenspaces of e; $H = \frac{e + id_H}{2}H \oplus \frac{id_H-e}{2}H$. Since $e$ commutes with $e_1,..,e_p$, we get a p-multigraded Hilbert space $\frac{e + id_H}{2}H$. This functor is also well-defined on morphisms, it just restricts to the eigenspace; $T \mapsto \frac{e' + id_{H'}}{2}T\frac{e + id_H}{2} \in Hom(FH,FH')$ for $T \in Hom(H,H')$ since T commutes with e.
1) any $\mathcal{C}_p$ linear map $T:\frac{e + id_H}{2}H \to \frac{e' + id_{H'}}{2}H'$ should be the restriction of some diagonal operator since it commutes with e, so is there anything to prove here?
2) Here i am in doubt, if two operators agree on the +1 eigenspaces, do they also agree on the complement?
3) is shown by taking a p-multigraded Hilbert space $(H, \{e_i\}_{i=1}^p)$ and considering the Hilbert space $H_1 = H \oplus H^{op}$. If $e_{p+1} = \begin{bmatrix} 0 & 1 \\ -1 & 0 \end{bmatrix}$ and $e_{p+2} = \begin{bmatrix} 0 & i \\ i & 0 \end{bmatrix}$ then $(H_1, \{e_i \oplus e_i^{op}\}_{i=1}^{p} \cup e_{p+1} \cup e_{p+2})$ is (p+2)-multigraded Hilbert space and $FH_1 \simeq H$.
So i am looking for some feedback on the above steps, and if im doing something completely wrong i would like to know.