On approximately orthogonality preserving and reversing operators

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.