Linear maps on L(ℓ_pⁿ,ℓ_p^m), (p∈ \1,∞\) preserving parallel pairs

2025/07/12 by ‎Arpita Mal, Mal, Arpita
Mathematics · #Advanced Algebra and Geometry #Advanced Differential Equations and Dynamical Systems #Advanced Topics in Algebra #FOS: Mathematics #Functional Analysis (math.FA)

paper · pdf · doi:10.48550/arxiv.2507.09284

openalex publication_date 2025/07/12 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Two vectors x,y of a Banach space are said to form a parallel (resp. triangle equality attaining or TEA) pair if ‖x+λy‖=‖x‖+‖y‖ holds for some scalar λ with |λ|=1 (resp. λ=1). For p∈ \1,∞\, and m,n≥ 2, we study the linear maps T: L(ℓ_pⁿ, ℓ_p^m) → L(ℓ_pⁿ,ℓ_p^m) that preserve parallel (resp. TEA) pairs, that is, those linear maps T for which T(A),T(B) form a parallel (resp. TEA) pair whenever A,B form a parallel (resp. TEA) pair of L(ℓ_pⁿ,ℓ_p^m). We prove that if T is non-zero, then the following are equivalent: (1) T preserves TEA pairs. (2) T preserves parallel pairs and rank(T)>1. (3) T preserves parallel pairs and T is invertible. (4) T is a scalar multiple of an isometry.

Citations

• Linear preservers of parallel matrix pairs with respect to the k-numerical radius
• Linear maps on matrices preserving parallel pairs
• Linear maps preserving parallel matrix pairs with respect to the Ky-Fan k-norm

Related

• Linear maps preserving ℓ_p-norm parallel vectors
• Linear maps on matrices preserving parallel pairs
• A study on preservation of parallel pairs and triangle equality attainment pairs
• Linear preservers of parallel matrix pairs with respect to the k-numerical radius
• Linear maps preserving parallel matrix pairs with respect to the Ky-Fan k-norm