An identity for (δ,ε)-approximately orthogonality preserving mappings

Suppose T is a Hilbert space operator. Given δ∈[0,1), we define εˆδ(T) to be the smallest ε for which T is (δ,ε)-approximately orthogonality preserving, and then obtain an exact formula for εˆδ(T) in terms of δ,‖T‖ and the minimum modulus m(T) of T. For two nonzero operators T,S, it follows from the formula that T is (εˆ(S),ε)-AOP if and only if S is (εˆ(T),ε)-AOP, where εˆ(T)=εˆ0(T). Finally, we show that an operator T is (δ,ε)-AOP if and only if there exists a “special” δ-AOP operator S such that TS is ε-AOP [Theorem 3.8].