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 $\lambda P+Q$ofapair$( P, Q)$ofprojectionsatsomepoint$\lambda\in$\mathbb{R}$\backslash{-1, 0}$\mathrm{.}Thenwerepresentallpairs$(P, Q)$ofprojectionssuchthat$T=\lambda P+Q$forafixed$\lambda$, and find that all such pairs are connected if $\lambda \in \mathbb{R}\mathrm{\backslash }\{-1,0,1\}$. 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.