Invertibility preserving linear maps and algebraic reflexivity of elementary operators of length one

Let X and Y be real or complex Banach spaces. We show that a surjective linear map phi : B (X) --> B (Y) preserving invertibility in both directions is either of the form phi (T) = ATB or the form phi (T) =CT'D, where A : X --> Y, B : Y --> X, C : X' --> Y, and D : Y --> X' are bounded invertible linear operators. As an application we improve a result of Larson and Sourour on algebraic reflexivity of elementary operators of length one.