Pencils of pairs of projections
Let $T$ be a self-adjoint operator on a complex Hilbert space $\mathcal{H}$. We give a sufficient and necessary condition for $T$ to be the pencil $\lambda P+Q$ of a pair $( P, Q)$ of projections at some point $\lambda\in\mathbb{R}\backslash\{-1, 0\}$. Then we represent all pairs $(P, Q)$ of projections such that $T=\lambda P+Q$ for a fixed $\lambda$, and find that all such pairs are connected if $\lambda\in\mathbb{R}\backslash\{-1, 0, 1\}$. Afterwards, the von Neumann algebra generated by such pairs $(P,Q)$ is characterized. Moreover, we prove that there are at most two real numbers such that $T$ is the pencils at these real numbers for some pairs of projections. Finally, we determine when the real number is unique.