Page 92 - Textos de Matemática Vol. 44

82 M. KARAMANLIS AND P. J. PSARRAKOS
χ ⊥ε ψ, if |⟨χ,ψ⟩| ≤ ε∥χ∥∥ψ∥. Furthermore, χ ⊥ ψ (resp., χ ⊥ε ψ) if and only if χ ⊥BJ ψ (resp., χ ⊥εBJ ψ) [4, 7].
Inspired by (1.1) and the above definition of Birkhoff-James ε-orthogonality, Chorianopoulos and Psarrakos [6] (see also [5] for a primer work) proposed the following definition for rectangular matrices: For any A, B ∈ Cn×m with B ̸= 0, any matrix norm ∥ · ∥, and any ε ∈ [0, 1), the Birkhoff-James ε-orthogonality set of A with respect to B is defined as
(1.2) Fε (A;B)={μ∈C:B⊥ε (A−μB)} ∥·∥ BJ
= μ∈C: ∥A−λB∥≥ 1−ε2∥B∥|μ−λ|, ∀λ∈C
∥A−λB∥ =Dλ,√ .
λ∈C
1 − ε2 ∥B∥
The Birkhoff-James ε-orthogonality set is a direct generalization of the stan-
dard numerical range. In particular, for n = m, ∥·∥ = ∥·∥2, B = In and
ε = 0, we have F0 (A;In) = F(A); see (1.1) and (1.2). Moreover, Fε (A;B) ∥·∥2 ∥·∥
is a non-empty, compact and convex subset of C that lies in the closed disk D 0, √ ∥A∥ and has interesting geometric properties [6].
1−ε2 ∥B∥
In this note, we adopt ideas and techniques from [6] to introduce and study
the Birkhoff-James ε-orthogonality set of elements of a complex normed linear space, generalizing results of [6]. In the next section, we give the definition of the set, and verify that it is always non-empty. In Section 3, we explore the growth of the set, and in Section 4, we derive characterizations of its interior and boundary. Finally, in Section 5, we describe the Birkhoff-James ε-orthogonality set when the norm is induced by an inner product.
2. The definition
Consider a complex normed linear space (X , ∥ · ∥) (for simplicity, X ), and let χ, ψ ∈ X with ψ ̸= 0. For any ε ∈ [0, 1), the Birkhoff-James ε-orthogonality set of χ with respect to ψ is defined and denoted by
(2.1) Fε (χ;ψ) = {μ∈C: ψ⊥ε (χ−μψ)}. ∥·∥ BJ
It is straightforward to see that