Trapezoid Orthogonality in Complex Normed Linear Spaces

Let G p ( x , y , z ) = ∥ x + y + z ∥ p + ∥ z ∥ p − ∥ x + z ∥ p − ∥ y + z ∥ p be defined on a normed space X . The special case G 2 ( x , y , z ) = 0 , ∀ z ∈ X , where X is a real normed linear space, coincides with the trapezoid orthogonality (T-orthogonality), which was originally proposed by Alsina et al. in 1999. In this paper, for the case where X is a complex inner product space endowed with the inner product ⟨ · , · ⟩ and induced norm ∥ · ∥ , it is proved that S g n ( G 2 ( x , y , z ) ) = S g n ( R e ⟨ x , y ⟩ ) , ∀ z ∈ X , and a geometric explanation for condition R e ⟨ x , y ⟩ = 0 is provided. Furthermore, a condition G 2 ( x , i y , z ) = 0 , ∀ z ∈ X is added to extend the T-orthogonality to the general complex normed linear spaces. Based on some characterizations, the T-orthogonality is compared with several other well-known types of orthogonality. The fact that T-orthogonality implies Roberts orthogonality is also revealed.