Maps preserving the ascent and descent of product of operators

Abstract

Let B(X) be the algebra of all bounded linear operators on a complex or real Banach space X with dim⁡X≥3. In this paper, we characterize the maps from B(X) into itself which preserve the ascent of product of operators or, they preserve the descent of product of operators. It turns out that both problems are connected with preservers of the rank-one nilpotency of the product.