The purpose of this paper is to present in linear spaces some results for new notions called A-left (resp., A-right) ascent and A-left (resp., A-right) descent of linear operators (where A is a given operator) which generalize two important notions in operator theory: ascent and descent. Moreover, if A is a positive operator, we obtain several properties of ascent and descent of an operator in semi-Hilbertian spaces. Some basic properties and many results related to the ascent and descent for a linear operator on a linear space Kaashoek (Math Ann 172:105–115, 1967), Taylor (Math Ann 163:18–49, 1966) are extended to these notions. Some stability results under perturbations by compact operators and operators having some finite rank power are also given for these notions.

本文的目的是在线性空间中呈现一些新概念的结果，称为线性算子的A -left (resp., A -right) 上升和A -left (resp., A -right) 下降（其中A是一个给定算子）概括了算子理论中的两个重要概念：上升和下降。此外，如果A是一个正算子，我们得到了一个算子在半希尔伯特空间中上升和下降的几个性质。与线性空间上线性算子的上升和下降相关的一些基本属性和许多结果 Kashoek (Math Ann 172:105–115, 1967), Taylor (Math Ann 163:18–49, 1966) 扩展到这些概念. 对于这些概念，还给出了在紧凑算子和具有有限秩幂的算子扰动下的一些稳定性结果。