AOP mappings and the distance to the scalar multiples of isometries

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