Page 99 - Textos de Matemática Vol. 44

BIRKHOFF-JAMES ε-ORTHOGONALITY SETS 89
Corollary 3.6. Let χ,ψ ∈ X with ψ ̸= 0. If χ is not a scalar multiple of ψ, then
1−1
C= Fn(χ;ψ).
∥·∥ n∈N
4. The interior and the boundary of Fε ∥·∥
(χ;ψ)
Consider the Birkhoff-James ε-orthogonality set Fε (χ;ψ), and denote its
∥·∥
interior by Int F ε (χ; ψ) , and its boundary by ∂F ε (χ; ψ). ∥·∥ ∥·∥
Proposition 4.1. Let χ,ψ ∈ X, with ψ ̸= 0. Then, for any ε ∈ [0,1), Int Fε (χ;ψ) ⊆ μ∈C: ∥χ−λψ∥> 1−ε2∥ψ∥|μ−λ|, ∀λ∈C .
Proof. If μ ∈ Int Fε (χ;ψ) , then there is a real ρ > 0 such that μ+ρeiθ ∈ ∥·∥
Fε (χ;ψ) for every θ ∈ [0,2π]. Hence, for every λ ∈ C, ∥·∥
∥χ−λψ∥ ≥ 1−ε2 ∥ψ∥|μ+ρeiθ −λ|, ∀θ∈[0,2π]. Setting θλ = arg(μ − λ), we observe that
∥χ−λψ∥ ≥ 1−ε2 ∥ψ∥ |μ+ρeiθλ −λ| > 1−ε2 ∥ψ∥ |μ−λ|,
completing the proof.
Theorem 4.2 (For matrices, see [6, Proposition 16]). Let χ, ψ ∈ X with ψ ̸= 0, and let ε ∈ [0,1). Suppose also that μ0 ∈ Fε (χ;ψ).
(i) The scalar μ0 lies on the boundary ∂Fε (χ;ψ) if and only if ∥·∥
inf ∥χ−λψ∥− 1−ε2 ∥ψ∥|μ0 −λ| = 0. λ∈C
(ii) If ε>0, then μ0 ∈∂Fε (χ;ψ) if and only if ∥·∥
min ∥χ−λψ∥− 1−ε2 ∥ψ∥|μ0 −λ| = 0,
λ∈C
orequivalently,ifandonlyif ∥χ−λ0ψ∥= 1−ε2∥ψ∥|μ0−λ0| for
some λ0 ∈ C.
Proof. (i) Suppose that μ0 is a boundary point of the Birkhoff-James
ε-orthogonality set (recall (2.3))
∥·∥
∥·∥
√
ε ∥χ−λψ∥
F∥·∥(χ;ψ)=
D λ,√1−ε2∥ψ∥ .
λ∈C