arXiv subjects
Publications and source records attributed to Qianglian Huang.
We solve the Yang-Baxter-like matrix equation $AXA = XAX$ for a general given matrix $A$ to get all anti-commuting solutions, by using the Jordan canonical form of $A$ and applying some new facts on a general homogeneous Sylvester equation. Our main result provides all the anti-commuting solutions of the nonlinear matrix equation.
In this note, we provide some sufficient and necessary conditions for the core inverse of the perturbed operator to have the simplest possible expression. The results improve the recent work by H. Ma (Optimal perturbation bounds for the core inverse, Appl. Math. Comput. 336 (2018) 176-181.).
In this paper, we establish a link between the Hyers-Ulam stability and the Moore--Penrose inverse, that is, a closed operator has the Hyers-Ulam stability if and only if it has a bounded Moore-Penrose inverse. Meanwhile, the stability constant can be determined in terms of the Moore-Penrose inverse. Based on this result, some conditions for the perturbed operators having the Hyers--Ulam stability are obtained and the Hyers-Ulam stability constant is expressed explicitly in the case of closed operators. In the case of the bounded linear operators we obtain some characterizations for the Hyers-Ulam stability constants to be continuous. As an application, we give a characterization for the Hyers-Ulam stability constants of the semi-Fredholm operators to be continuous.
Utilizing the stability characterizations of generalized inverses of linear operator, we investigate the existence of generalized resolvents of linear pencils in Banach spaces. Some practical criterions for the existence of generalized resolvents of the linear pencil $λ\rightarrow T-λS$ are provided and an explicit expression of the generalized resolvent is given. As applications, the characterization for the Moore-Penrose inverse of the linear pencil to be its generalized resolvent and the existence of the generalized resolvents of linear pencils of finite rank operators, Fredholm operators and semi-Fredholm operators are also considered. The results obtained in this paper extend and improve many results in this area.