Trapezoid Orthogonality in Complex Normed Linear Spaces

Abstract

Let Gp(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 G2(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 Sgn(G2(x,y,z))=Sgn(Re⟨x,y⟩),∀z∈X, and a geometric explanation for condition Re⟨x,y⟩=0 is provided. Furthermore, a condition G2(x,iy,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.

Keywords

• trapezoid orthogonality (T-orthogonality)
• complex normed linear space
• complex inner product space
• Roberts orthogonality
• Birkhoff orthogonality