Page 91 - Textos de Matemática Vol. 44

BIRKHOFF-JAMES ε-ORTHOGONALITY SETS IN NORMED LINEAR SPACES
MILTIADIS KARAMANLIS AND PANAYIOTIS J. PSARRAKOS
Dedicated to Professor Nata´lia Bebiano on the occasion of her 60th birthday
Abstract. Consider a complex normed linear space (X,∥ · ∥), and let χ,ψ ∈ X with ψ ̸= 0. Motivated by a recent work of Chorianopoulos and Psarrakos (2011) on rectangular matrices, we introduce the Birkhoff-James
ε-orthogonality set of χ with respect to ψ, and explore its rich structure.
1. Introduction
The numerical range (also known as the field of values) of a square complex matrixA∈Cn×n isdefinedasF(A)={x∗Ax∈C: x∈Cn, x∗x=1} [8].This range is a non-empty, compact and convex subset of C, which has been studied extensively and is useful in understanding matrices and operators; see [2, 3, 8, 10] and the references therein. The numerical range F(A) is also written in theform(see[3,10])F(A)={μ∈C: ∥A−λIn∥2 ≥|μ−λ|, ∀λ∈C},where ∥ · ∥2 denotes the spectral matrix norm (i.e., that norm subordinate to the euclidean vector norm) and In is the n × n identity matrix. As a consequence, F(A) is an infinite intersection of closed (circular) disks D(λ,∥A−λIn∥2) = {μ∈C: |μ−λ|≤∥A−λIn∥2}(λ∈C),namely,
(1.1) F(A) = {μ∈C: |μ−λ|≤∥A−λIn∥2} = D(λ,∥A−λIn∥2). λ∈C λ∈C
For two elements χ and ψ of a complex normed linear space (X , ∥ · ∥),
χ is said to be Birkhoff-James orthogonal to ψ, denoted by χ ⊥BJ ψ, if
∥χ + λψ∥ ≥ ∥χ∥ for all λ ∈ C [1, 9]. This orthogonality is homogeneous,
but it is neither symmetric nor additive [9]. Moreover, for any ε ∈ [0, 1), χ is
called Birkhoff-James ε-orthogonal to ψ, denoted by χ ⊥ε ψ, if ∥χ + λψ∥ ≥
√
1 − ε2 ∥χ∥ for all λ ∈ C [4, 7]. It is worth mentioning that this relation is
also homogeneous. In an inner product space (X,⟨·,·⟩), with the standard or- thogonality relation ⊥ , a χ ∈ X is called ε-orthogonal to a ψ ∈ X , denoted by
2010 Mathematics Subject Classification. 46B99, 47A12
Key words and phrases. norm, Birkhoff-James orthogonality, Birkhoff-James ε-orthogonality, numerical range, inner product
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