Pencils of pairs of projections

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Let T be a self-adjoint operator on a complex Hilbert space H. We give a sufficient and necessary condition for T to be the pencil λP+Q of a pair (P, Q) of projections at some point λ{-1, 0}. Then we represent all pairs (P, Q) of projections such that T=λP+Q for a fixed λ, and find that all such pairs are connected if λ{-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.