Geometric Characterization of the Numerical Range of Parallel Sum of Two Orthogonal Projections

Abstract

Let $\mathcal{H}$ be a complex separable Hilbert space and $\mathcal{B}\left(\mathcal{H}\right)$ be the algebra of all bounded linear operators from $\mathcal{H}$ to $\mathcal{H}$. Our goal in this article is to describe the closure of numerical range of parallel sum operator $P:\text{PQ}$ for two orthogonal projections $P$ and $Q$ in $\mathcal{B}\left(\mathcal{H}\right)$ as a closed convex hull of some explicit ellipses parameterized by points in the spectrum.