Ascent and descent of bounded linear operators

Let $\mathcal B(\mathcal X)$ be the algebra of all bounded linear operators on a real or complex Banach space $\mathcal{X}$ with $\dim\mathcal X \ge 3$.

In this paper, we first explore the ascent (descent) of upper triangular block operator matrices and certain special algebraic operators, and then establish characterizations for the ascent (descent) of rank-one and rank-two operators.

Based on these results, we characterize features for some special operators by the ascent (descent) of Jordan products.

As an application, we give the structure of all maps with range containing all bounded operators of rank at most three preserving the ascent (descent) of operator Jordan product on $\mathcal B(\mathcal X)$.