Trapezoid Orthogonality in Complex Normed Linear Spaces

Zheng Li (), Tie Zhang and Changjun Li
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Zheng Li: Department of Mathematics, Northeastern University, Shenyang 110004, China
Tie Zhang: Department of Mathematics, Northeastern University, Shenyang 110004, China
Changjun Li: School of Computer and Software Engineering, University of Science and Technology Liaoning, Anshan 114051, China

Mathematics, 2025, vol. 13, issue 9, 1-14

Abstract: 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.

Keywords: trapezoid orthogonality (T-orthogonality); complex normed linear space; complex inner product space; Roberts orthogonality; Birkhoff orthogonality (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2025
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