Maps preserving ascent/descent of triple Jordan product

Abstract

Let $\mathcal{X}$ be a real or complex Banach space with $\dim \mathcal{X}\ge 3$. We give a complete description of surjective mappings on $\mathcal{B}(\mathcal{X})$ that preserve the ascent of Jordan triple product of operators or, preserve the descent of Jordan triple product of operators.