Spectrum-preserving additive maps

AbstractLet X and Y be complex Banach spaces. We show that a spectrum-preserving surjective additive map Φ from B(X) to B(Y) is either of the form Φ(F)=ATA-1 for a linear isomorphism A of X onto Y or of the form Φ(T)=BT∗B-1 for a linear isomorphism B of X∗ onto Y