AI 中文总结
研究实或复巴拿赫空间上有界线性算子的升阶与降阶，通过探究特殊算子的升阶降阶，建立相关刻画，进而通过约旦积的升阶降阶刻画特殊算子特征，并给出保持算子约旦积升阶降阶的映射结构。
AI 中文摘要
设\(\mathcal B(\mathcal X)\)为实或复巴拿赫空间\(\mathcal{X}\)（\(\dim\mathcal X \ge 3\)）上所有有界线性算子构成的代数。本文首先探究上三角块算子矩阵及某些特殊代数算子的升阶（降阶），接着建立秩一和秩二算子升阶（降阶）的刻画。基于这些结果，通过约旦积的升阶（降阶）刻画一些特殊算子的特征。作为应用，给出值域包含所有秩至多为三的有界算子且保持\(\mathcal B(\mathcal X)\)上算子约旦积升阶（降阶）的所有映射的结构。
英文摘要
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 $\mathcal B(\mathcal X)$.
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