Yifeng Xue

Jianbing Cao, Yifeng Xue

Let $X, Y$ be Banach spaces and $T : X \to Y$ be a bounded linear operator. In this paper, we initiate the study of the perturbation problems for bounded homogeneous generalized inverse $T^h$ and quasi--linear projector generalized inverse $T^H$ of $T$. Some applications to the representations and perturbations of the Moore--Penrose metric generalized inverse $T^M$ of $T$ are also given. The obtained results in this paper extend some well--known results for linear operator generalized inverses in this field.

Jianbing Cao, Yifeng Xue

In this paper, we first study the perturbations and expressions for the generalized inverses $a^{(2)}_{p,q}$, $a^{(1, 2)}_{p,q}$, $a^{(2, l)}_{p,q}$ and $a^{(l)}_{p,q}$ with prescribed idempotents $p$ and $q$. Then, we investigate the general perturbation analysis and error estimate for some of these generalized inverses when $p,\,q$ and $a$ also have some small perturbations.

Fapeng Du, Yifeng Xue

In this paper, we investigate the invertibility of $I_Y+δTT^+$ when $T$ is a closed operator from $X$ to $Y$ with a generalized inverse $T^+$ and $δT$ is a linear operator whose domain contains $D(T)$ and range is contained in $D(T^+)$. The characterizations of the stable perturbation $T+δT$ of $T$ by $δT$ in Banach spaces are obtained. The results extend the recent main results of Huang's in Linear Algebra and its Applications.

Fapeng Du, Yifeng Xue

Let $\R $ be a ring with unit 1 and $a\in \R, \bar{a}=a+δa\in \R $ such that $a^#$ exists. In this paper, we mainly investigate the perturbation of the group inverse $a^#$ on $\R$. Under the stable perturbation, we obtain the explicit expressions of $\bar{a}^#$. The results extend the main results in Xue (2007), and Xue and Chen (2007) and some related results in Xue (2012). As an application, we give the representation of the group inverse of the matrix d&b c&0 on the ring $\R$ for certain $d, b, c\in\R$.

Fapeng Du, Yifeng Xue

In this paper, the perturbation problems of $A_{T,S}^{(2)}$ are considered. By virtue of the gap between subspaces, we derive the conditions that make the perturbation of $A_{T,S}^{(2)}$ is stable when $T,S$ and $A$ have suitable perturbations. At the same time, the explicit formulas for perturbation of $A_{T,S}^{(2)}$ and new results on perturbation bounds are obtained.

Fapeng Du, Yifeng Xue

In this paper, we investigate the perturbation for the Moore-Penrose inverse of closed operators on Hilbert spaces. By virtue of a new inner product defined on $H$, we give the expression of the Moore-Penrose inverse $\bar{T}^†$ and the upper bounds of $\|\bar{T}^†\|$ and $\|\bar{T}^†-T^†\|$. These results obtained in this paper extend and improve many related results in this area.

Jianbing Cao, Yifeng Xue

In this paper, the problems of perturbation and expression for the Moore--Penrose metric generalized inverses of bounded linear operators on Banach spaces are further studied. By means of certain geometric assumptions of Banach spaces, we first give some equivalent conditions for the Moore--Penrose metric generalized inverse of perturbed operator to have the simplest expression $T^M(I+ δTT^M)^{-1}$. Then, as an application our results, we investigate the stability of some operator equations in Banach spaces under different type perturbations.

Fapeng Du, Yifeng Xue

In this paper, we investigate the perturbation analysis of $A_{T,S}^{(2)}$ when $T,\,S$ and $A$ have some small perturbations on Hilbert spaces. We present the conditions that make the perturbation of $A_{T,S}^{(2)}$ is stable. The explicit representation for the perturbation of $A_{T,S}^{(2)}$ and the perturbation bounds are also obtained.