Maps preserving semi-Fredholm operators on Hilbert -modules

Let M be a Hilbert C ∗ -module over the C ∗ -algebra A , and L A ( M ) the C ∗ -algebra of all adjointable operators on M . Let φ : L A ( M ) → L A ( M ) be a unital linear map preserving semi- A -Fredholm operators in both directions and assume that φ is surjective up to compact operators. We will show that φ preserves compact operators and in certain cases, the induced map φ ˜ : L A ( M ) / K A ( M ) → L A ( M ) / K A ( M ) is an automorphism or an anti-automorphism.