On approximately orthogonality preserving and reversing operators

TL;DR

This paper investigates operators that approximately preserve or reverse orthogonality in Banach spaces, characterizing their structure and showing their relation to injectivity and scalar multiples of approximate isometries.

Contribution

It provides new characterizations of approximately orthogonality preserving/reversing operators, linking them to injectivity and scalar multiples of ε-isometries in finite-dimensional Banach spaces.

Findings

Injective operators are approximately orthogonality preserving/reversing.

Operators between n-dimensional Banach spaces are scalar multiples of ε-isometries.

Maps close to ε-isometries are approximately orthogonality preserving/reversing.

Abstract

We study approximately orthogonality (in the sense of Dragomir) preserving and reversing operators. We show that for some orthogonality notations, an operator defined from a finite-dimensional Banach space to a normed linear space is approximately orthogonality preserving/reversing if and only if it is an injective operator. This result implies that for some orthogonality notations, any operator defined from an $n$-dimensional Banach space to another $n$-dimensional Banach space is approximately orthogonality preserving/reversing if and only if it is a scalar multiple of an $\varepsilon$-isometry. We show that any $\varepsilon$-isometry and maps close to $\varepsilon$-isometries defined from a normed linear space to another normed linear space are approximately orthogonality preserving/reversing for some orthogonality notations. We also study the locally approximate orthogonality preserving and reversing operators defined on some finite-dimensional Banach spaces.