Work

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

Year: 2026

Type: article

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.

Source: Mathematics

DOI: 10.3390/math14101732

Authors: Liangyu Fu, Ran Wang, Weiyan Yu

Institution: Hainan Normal University

Language: English

Topic: Holomorphic and Operator Theory

Subfield: Applied Mathematics

Field: Mathematics

Domain: Physical Sciences

Open Access status: gold

Estimated APC paid: $3,137

Funder: National Natural Science Foundation of China