A note on linear approximately orthogonality preserving mappings

In this paper, linear $\epsilon$-orthogonality preserving mappings are studied. We define $\widehat{\epsilon }\left(T\right)$ as the smallest $\epsilon$ for which $T$ is $\epsilon$-orthogonality preserving, and then derive an exact formula for $\widehat{\epsilon }\left(T\right)$ in terms of $\parallel T\parallel$ and the minimum modulus $m\left(T\right)$ of $T$. We see that $\epsilon$-orthogonality preserving mappings (for some $\epsilon <1$) 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.