A note on linear approximately orthogonality preserving mappings

TL;DR

This paper investigates linear mappings that approximately preserve orthogonality, providing an exact formula for their approximation measure and linking them to bounded below operators, while improving existing stability bounds.

Contribution

It derives an exact formula for the smallest epsilon making a linear map epsilon-orthogonality preserving and refines stability bounds in the context.

Findings

Exact formula for -hat() in terms of operator norm and minimum modulus

Characterization of --preserving maps as bounded below operators

Improved upper bound in the stability equation from previous work

Abstract

In this paper, linear $\varepsilon$-orthogonality preserving mappings are studied. We define $\hat{\varepsilon}\left(T\right) $ as the smallest $\varepsilon$ for which $T$ is $\varepsilon$-orthogonality preserving, and then derive an exact formula for $\hat{\varepsilon}\left( T\right) $ in terms of $\left \Vert T\right \Vert $ and the minimum modulus $m\left( T\right) $ of $T$. We see that $\varepsilon$-orthogonality preserving mappings (for some $\varepsilon<1$) are exactly the operators that are bounded from below. We improve an upper bounded in the stability equation given in [7, Theorem 2.3], which was thought to be sharp.