Maps preserving the nilpotency of products of operators

Let B (X) be the algebra of all bounded linear operators on the Banach space X, and let N (X) be the set of nilpotent operators in B (X). Suppose φ{symbol} : B (X) → B (X) is a surjective map such that A, B ∈ B (X) satisfy AB ∈ N (X) if and only if φ{symbol} (A) φ{symbol} (B) ∈ N (X). If X is infinite dimensional, then there exists a map f : B (X) → C {minus 45 degree rule} {0} such that one of the following holds:(a)There is a bijective bounded linear or conjugate-linear operator S : X → X such that φ{symbol} has the form A {mapping} S [f (A) A] S- 1.(b)The space X is reflexive, and there exists a bijective bounded linear or conjugate-linear operator S : X′ → X such that φ{symbol} has the form A {mapping} S[f(A)A′]S-1. If X has dimension n with 3 ≤ n < ∞, and B (X) is identified with the algebra Mnof n × n complex matrices, then there exist a map f : Mn→ C {minus 45 degree rule} {0}, a field automorphism ξ : C → C, and an invertible S ∈ Mnsuch that φ{symbol} has one of the following forms:A = [aij] {mapping} f (A) S [ξ (aij)] S- 1or A = [aij] {mapping} f (A) S [ξ (aij)]tS- 1,where Atdenotes the transpose of A. The results are extended to the product of more than two operators and to other types of products on B (X) including the Jordan triple product A * B = ABA. Furthermore, the results in the finite dimensional case are used to characterize surjective maps on matrices preserving the spectral radius of products of matrices.