Let A and B be standard operator algebras on Banach spaces X and Y, respectively. Let ϕ : A → B be a bijective map. In this paper, we show that ϕ is completely preserving quadratic operator in both directions if and only if ϕ is 2-quadratic preserving operator in both directions and if and only if ϕ is either an isomorphism or (in the complex case) a conjugate isomorphism.
