Page 94 - Textos de Matemática Vol. 44

84 M. KARAMANLIS AND P. J. PSARRAKOS
Proof. By the defining formula (2.1) of the Birkhoff-James ε-orthogonality set F ε (χ; ψ) and the homogeneity of the Birkhoff-James ε-orthogonality, it is
straightforwardthatFε (χ;bψ)={μ∈C: ψ⊥ε (χ−(bμ)ψ)}. ∥·∥ BJ
Proposition 2.3. Let χ and ψ be two nonzero elements of X. Then, for any ε ∈ [0, 1),
−1 ε ∥χ∥ ε
μ ∈ C : μ ∈ F∥·∥(χ;ψ), |μ| ≥ ∥ψ∥ ⊆ F∥·∥(ψ;χ).
|λ| ψ−λχ ≥ 1−ε2∥ψ∥|λ| λ−1 , ∀λ∈C\{0},
∥·∥
Proof. Consider a μ ∈ F ε (χ; ψ) with |μ| ≥ ∥χ∥ . Then, by (2.2), we have
∥·∥
∥ψ∥ 1 μ
or
∥ψ−λχ∥ ≥ 1−ε2∥ψ∥|μ| μ −λ ≥ 1−ε2∥χ∥ μ −λ , ∀λ∈C.
−1 −1
Thus, μ−1 lies in Fε (ψ;χ).
∥·∥
Proposition 2.4. Let ∥ · ∥a and ∥ · ∥b be two equivalent norms acting in X , and suppose that for two real numbers C, c > 0, c ∥ζ∥a ≤ ∥ζ∥b ≤ C ∥ζ∥a for allζ∈X.Then,foranyχ,ψ∈X withψ̸=0andanyε∈[0,1),itholdsthat
where ε′ = 1−c2(1−ε2). C2
Fε (χ;ψ) ⊆ Fε′ (χ;ψ), ∥·∥a ∥·∥b
Proof. Suppose μ ∈ F ε (χ; ψ). Then, it follows readily that ∥·∥a
or or
11+i Forexample,weconsiderthevectors χ= 2+i ,ψ= 2+i ∈C3,
−11i i
∥χ−λψ∥a ≥ 1−ε2∥ψ∥a|μ−λ|, ∀λ∈C, ∥χ−λψ∥b ≥ 1−ε2 c ∥ψ∥b|μ−λ|, ∀λ∈C,
C
c2(1−ε2) 2
∥χ−λψ∥b ≥ 1− 1− C2 ∥ψ∥b|μ−λ|, ∀λ∈C, and the proof is complete.

and recall that the (equivalent in C3) norms ∥·∥2 and ∥·∥1 satisfy ∥ζ∥2 ≤ ∥ζ∥1 ≤
√3∥ζ∥2 for all ζ ∈ C3. The Birkhoff-James ε-orthogonality sets F0.5 (χ;ψ), ∥·∥2