On a rho -Orthogonally Additive Mappings

Jacek Chmieliński,Paweł Wójcik,Justyna Sikorska

doi:10.1007/s00025-020-01238-9

Jacek Chmieliński, Paweł Wójcik + Show 1 more

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https://doi.org/10.1007/s00025-020-01238-9

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Abstract

We show that a real normed linear space endowed with the rho -orthogonality relation, in general need not be an orthogonality space in the sense of Rätz. However, we prove that rho -orthogonally additive mappings defined on some classical Banach spaces have to be additive. Moreover, additivity (and approximate additivity) under the condition of an approximate orthogonality is considered.

Highlights

Inner product spaces are by all means the most natural venue for orthogonality, allowing the definition: x⊥y ⇔ x|y = 0
In a real normed linear space (X, · ), for two vectors x, y ∈ X, one can consider for example the Birkhoff-James orthogonality ⊥B defined by x⊥By ⇐⇒ ∀ λ ∈ R x ≤ x + λy, or the isosceles orthogonality ⊥i defined by x⊥iy ⇐⇒ x + y = x − y, or many others
It is seen that any inner product space is an orthogonality space in the above sense

Summary

Introduction

Analogous relations may be considered in normed linear spaces, as well as in more general settings. In a real normed linear space (X, · ), for two vectors x, y ∈ X, one can consider for example the Birkhoff-James orthogonality ⊥B (see [2, 15]) defined by x⊥By ⇐⇒ ∀ λ ∈ R x ≤ x + λy , or the isosceles orthogonality ⊥i (see [15]) defined by x⊥iy ⇐⇒ x + y = x − y , or many others. Some axiomatic definitions of the orthogonality in linear spaces (or even more general structures) are known, among them the one formulated by Ratz [18] (compare with [12]).

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