Maps preserving the nilpotency of products of operators

Abstract

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 : B ( X ) → B ( X ) is a surjective map such that A , B ∈ B ( X ) satisfy AB ∈ N ( X ) if and only if ϕ ( A ) ϕ ( B ) ∈ 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 ϕ has the form A ↦ 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 has the form A ↦ S[ f( A) A′] S −1. If X has dimension n with 3 ⩽ n < ∞, and B ( X ) is identified with the algebra M n of n × n complex matrices, then there exist a map f : M n → C ⧹ { 0 } , a field automorphism ξ : C → C , and an invertible S ∈ M n such that ϕ has one of the following forms: A = [ a ij ] ↦ f ( A ) S [ ξ ( a ij ) ] S - 1 or A = [ a ij ] ↦ f ( A ) S [ ξ ( a ij ) ] t S - 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 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.