Since for projectors $\lbrace P_\alpha \rbrace_{\alpha} $ and $\lbrace Q_\beta \rbrace_{\beta} $ on a hilbert space $\mathcal{H}$ that are each a partition of unity:
$$\sum_\alpha P_\alpha Q_\beta = Q_\beta$$ $$\sum_\beta P_\alpha Q_\beta = P_\alpha$$
is there a function $f:\mathbb{C} \rightarrow \mathbb{C}$ and a function $g:\mathbb{C} \rightarrow \mathbb{C}$ such that their functional calculus is such that:
$$f(P_\alpha Q_\beta) = \sum_\alpha P_\alpha Q_\beta = Q_\beta$$ $$g(P_\alpha Q_\beta) = \sum_\beta P_\alpha Q_\beta = P_\alpha$$
If so, which functions are these? I am trying to prove the easier version of the proposition that for every element $M$ of a context $\mathcal{C}$ there is a generating element of this context $\widetilde{M}$ and a real function $f:\mathbb{R} \rightarrow \mathbb{R}$ such that: $f(\widetilde{M})= M$. But considering only projectors, and not making mention to this generating element the context $\widetilde{M}$.