Maps preserving the nilpotency of products of operators Original Research Article

Let image be the algebra of all bounded linear operators on the Banach space X, and let image be the set of nilpotent operators in image. Suppose image is a surjective map such that image satisfy image if and only if image. If X is infinite dimensional, then there exists a map image 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 Amaps toS[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 maps to S[f(A)A′]S−1. If X has dimension n with 3 less-than-or-equals, slant n < ∞, and image is identified with the algebra Mn of n × n complex matrices, then there exist a map image, a field automorphism image, and an invertible S set membership, variant Mn such that phi has one of the following forms:imagewhere At denotes the transpose of A. The results are extended to the product of more than two operators and to other types of products on image 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.