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May 20, 2026Mathematics0 citationsOpen Access

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

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LFLiangyu FuRWRan WangWYWeiyan Yu

Key Points

  • This research aims to explore the numerical ranges of generalized pencil operators formed by pairs of orthogonal projections.
  • Studied the closure of the numerical range for the operator T=P+αQ+βPQ.
  • Utilized Halmos’ two-subspace theorem to derive properties of the numerical range.
  • Analyzed the relationship between the spectrum and numerical range of the operators.
  • The closed convex hull of the numerical range is identified as a family of parametrized ellipses E(λ).
  • The spectrum σ(T) aligns with the foci of the elliptic family, demonstrating geometric connections.
  • Suitable assumptions permit this characterization, expanding understanding of numerical ranges.

Abstract

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.

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Cite This Study

Fu et al. (2026) studied this question.

synapsesocial.com/papers/6a0d4f62f03e14405aa9aa7chttps://doi.org/10.3390/math14101732
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