Nonlinear maps preserving certain subspaces.
Keywords:
Kernel operator, Nonlinear preservers problem, Range operatorAbstract
Let X be a Banach space and let B(X) be the Banach algebra of all bounded linear operators on X. We characterise surjective (not necessarily linear or additive) maps ϕ : B(X) → B(X) such that F(ϕ (A)◇ ϕ (B)) = F(A ◇ B) for all A,B ∈ B(X) where F(A) denotes any of R(A) or N(A), anda ◇ B denotes any binary operations A−B, AB and ABA for all A,B ∈B(X).References
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