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 $\phi:{\mathcal B}(X) \to {\mathcal B}(Y)$ preserving invertibility in both directions is either of the form $\phi(T)=ATB$ or the form $\phi(T)=CT'D$, where $A:X\to Y$, $B:Y\to X$, $C:X' \to Y$, and $D:Y\to X'$ 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.