Pencils of pairs of projections
Volume 249 / 2019
Abstract
Let $T$ be a self-adjoint operator on a complex Hilbert space $\mathcal {H}$. In this paper, a sufficient and necessary condition for $T$ to be (the value of) the pencil $\lambda P+Q$ of a pair $( P, Q)$ of projections at some point $\lambda \in \mathbb {R}\backslash \{-1, 0\}$ is introduced. Then we give a representation of all pairs $(P, Q)$ of projections such that $T=\lambda P+Q$ for a fixed real number $\lambda $, and find that all such pairs constitute a connected set if $\lambda \in \mathbb {R}\backslash \{-1, 0, 1\}$. Further, the von Neumann algebra generated by such pairs $(P,Q)$ is characterized. Moreover, we prove that there are at most two non-zero real numbers such that $T$ is the pencil of a pair of projections at these numbers. Finally, we determine when there is only one such number.
