Page 97 - Textos de Matemática Vol. 44

∥·∥
BIRKHOFF-JAMES ε-ORTHOGONALITY SETS 87
For the converse, suppose that F ε (χ; ψ) = {a} for an ε ∈ (0, 1). Then, by ∥·∥
Corollary 3.2, χ is a scalar multiple of ψ, i.e., there is a b ∈ C such that χ = bψ. As a consequence,
|a−λ| ≥ 1−ε2|b−λ|, ∀λ∈C.
For λ = a, it follows that |b − a| = 0, and the proof is complete.
By the proof of the previous proposition, it is clear that if F ε (χ; ψ) = {a} ∥·∥
for an ε ∈ (0,1), then χ = aψ, and consequently, Fε (χ;ψ) = {a} for all ∥·∥
ε ∈ [0, 1).
Proposition 3.4. Let χ,ψ ∈ X with ψ ̸= 0, and let ε ∈ [0,1). Then, for any
a,b∈C, Fε (aχ+bψ;ψ)=aFε (χ;ψ)+b. ∥·∥ ∥·∥
Proof. If a = 0, then Proposition 3.3 yields Fε (aχ;ψ) = {0} = 0Fε (ψ;ψ). ∥·∥ ∥·∥
If a ̸= 0, then
Fε (aχ;ψ)= μ∈C: ∥aχ−λψ∥≥ 1−ε2∥ψ∥|μ−λ|, ∀λ∈C
λ μλ = μ∈C: χ−aψ ≥ 1−ε2∥ψ∥ a−a ,∀λ∈C

2 μ
= μ∈C:∥χ−λψ∥≥ 1−ε ∥ψ∥ a−λ ,∀λ∈C =aFε (χ;ψ).
∥·∥
∥·∥
Furthermore, for any a, b ∈ C,
Fε (aχ+bψ;ψ)= μ∈C:∥aχ+(b−λ)ψ∥≥ 1−ε2∥ψ∥|μ−λ|, ∀λ∈C
= μ∈C:∥aχ−λψ∥≥ 1−ε2∥ψ∥|(μ−b)−λ|, ∀λ∈C = μ ∈ C : μ−b ∈ Fε (aχ;ψ) ,
∥·∥
and the proof is complete. If we allow the value ε = 1, then (2.2) implies that F 1 (A; B) = C. Further-
more, if χ is not a scalar multiple of ψ, then Fε (A;B) can be arbitrarily large ∥·∥
for ε sufficiently close to 1.
Theorem 3.5 (For matrices, see [6, Proposition 4]). Let χ, ψ ∈ X with ψ ̸= 0,
and suppose that χ is not a scalar multiple of ψ. Then, for any bounded region
Ω⊂C,thereisanεΩ ∈[0,1)suchthatΩ⊆FεΩ(χ;ψ). ∥·∥
∥·∥