An orthogonality relation in complex normed spaces based on norm derivatives

2022/04/24 by S. M. Enderami, Enderami, S. M., M. Abtahi +5 · 1 citation
Computer Science · Mathematics · #47B49 #FOS: Mathematics #Fixed Point Theorems Analysis #Functional Analysis (math.FA) #Functional Equations Stability Results #Optimization and Variational Analysis #Primary 46B20 #Secondary 46C50

paper · pdf · doi:10.48550/arxiv.2205.06246

openalex publication_date 2022/04/24 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let X be a complex normed space. Based on the right norm derivative ρ_₊, we define a mapping ρ__∞ by ρ__∞(x,y) = \frac1π∫₀^2πe^iθρ_₊(x,e^iθy)dθ (x,y∈ X). The mapping ρ__∞ has a good response to some geometrical properties of X. For instance, we prove that ρ__∞(x,y)=ρ__∞(y,x) for all x, y ∈ X if and only if X is an inner product space. In addition, we define a ρ__∞-orthogonality in X and show that a linear mapping preserving ρ__∞-orthogonality has to be a scalar multiple of an isometry. A number of challenging problems in the geometry of complex normed spaces are also discussed.

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• Subdifferential of the B(H,K) norm, and approximate orthogonality

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