Maps preserving the nilpotency of products of operators

Let M(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 phi : B(X) -> B(X) is a surjective map such that A, B is an element of B(X) satisfy A B is an element of N(X) if and only if phi(A)phi(B) is an element of N(X). If X is infinite dimensional, then there exists a map f : B(X) -> C \ {0} such that one of the following holds: (a) There is a bijective bounded linear or conjugate-linear operator S : X -> X such that phi has the form A (bar right arrow) 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 phi has the form A (bra right arrow) S[f(A)A']S-1. If X has dimension n with 3 <= n < infinity, and B(X) is identified with the algebra M-n of n x n complex matrices, then there exist a map f : M-n -> C \ {0}, a field automorphism xi : C -> C, and an invertible S is an element of M-n such that 0 has one of the following forms: A = [a(ij)] (bar right arrow) f(A)S[xi(a(ij))]S-1 or A = [a(ij)] (bar right arrow) f(A)S[xi(a(ij))]S-t(-1), where A(t) denotes 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 Jorldan 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. (C) 2006 Elsevier Inc. All rights reserved.