Maps preserving the nilpotency of products of operators (Q884421)

This paper presents contributions to the study of nonlinear preservers. Let \(X\) be a complex Banach space with \(\dim X\geq 3\) and denote by \(B(X)\) the algebra of all bounded linear operators on \(X\). The authors give a complete description of the structure of all surjective maps \(\phi:B(X)\to B(X)\) (without assuming linearity) which have the property that, for every \(A,B\in B(X)\), \[ AB\text{ is nilpotent}\Longleftrightarrow \phi(A)\phi(B) \text{ is nilpotent}. \] Next, the results are extended to the product of more than two operators and to other types of products, including the Jordan triple product \(A*B=ABA\). Moreover, the results in the finite-dimensional case are used to characterize surjective maps on matrix algebras preserving the spectral radius of products of matrices.