Abstract:

Let B(X) be the Banach algebra of all bounded linear operators on an infinite dimensional complex Banach space X. We consider additive surjective maps preserving operators range or kernel inclusion in both directions. It is proved that if φ is an additive surjective map on B(X) preserving operators range or kernel inclusion in both directions, then there exist invertible bounded linear or conjugate linear operators U and V on X such that for every T∈B(X), φ(T)=UTV.

Key words: inclusion, range, kernel, additive maps