Linear maps preserving the set of Fredholm operators

Let $H$ be an infinite-dimensional separable complex Hilbert space and $\mathcal{B}(H)$ the algebra of all bounded linear operators on $H$. In this paper we characterize surjective linear maps $\phi : F\mathcal{B}(H)\to \mathcal{B}(H)$ preserving the set of Fredholm operators in both directions. As an application we prove that $\phi$ preserves the essential spectrum if and only if the ideal of all compact operators is invariant under $\phi$ and the induced linear map $\varphi$ on the Calkin algebra is either an automorphism, or an anti-automorphism. Moreover, we have, either $ind(\phi(T)) = ind(T)$ or $ind(\phi(T)) = - ind(T)$ for every Fredholm operator $T$.