Geometric Characterization of the Numerical Ranges of Generalized Pencils of Pairs of Projections

Let H be a complex separable Hilbert space. We study the closure of the numerical range of the generalized pencil T=P+αQ+βPQ, where (P,Q) is a pair of orthogonal projections and (α,β)∈R2. Using Halmos’ two-subspace theorem, it is shown that, under suitable assumptions, W(T)¯ is the closed convex hull of a family of ellipses E(λ) parametrized by λ∈σ(PQ). Moreover, the spectrum σ(T) coincides with the set of all foci of this elliptic family, revealing a precise geometric relation between the spectrum and the numerical range of such operators.