Banach空间中集值度量广义逆齐性单值选择的判据

本文研究了Banach空间(X,‖·‖),(Y,‖·‖)上具有闭值域的稠定闭算子T:X→Y的(集值)度量广义逆.在限定X为自反的、Y为一般的Banach空间且算子值域R(T)为空间Y中Chebyshev子空间时,证明了算子T具有非空闭凸集值的度量广义逆的存在性,运用Banach空间中广义正交分解定理,得出算子T的集值度量广义逆具有唯一齐性单值选择,并且该单值选择恰为赋等价严格凸范数的空间X_(r)=(X,‖·‖_(r))上算子T的Moore-Penrose度量广义逆.特别地,将抽象的Banach空间X与Y具体化为有限维Banach空间l_(1)^(n)=(R^(n),‖·‖1)(即n维空间R^(n)赋l1范数)与有限维Hilbert空间(即m维欧式空间l_(2)^(m)=(R^(m),‖·‖_(2)),亦即m维空间赋l_(2)范数),线性算子T可具体表示为m×n阶矩阵A,得到了从n维空间l_(1)^(n)到m维空间l_(2)^(m)有界线性算子A的集值度量广义逆的线性单值选择恰为A的Moore-Penrose逆A^(+).本文的工作响应了Nashed与Votruba在[Bull.Amer.Math.Soc.,1974,80(5):831-835]中提出的“如何获得线性和非线性算子度量广义逆具有良好性质的选择值得研究”的建议.

Perturbations of Moore–Penrose Metric Generalized Inverses of Linear Operators in Banach Spaces

In this paper, the perturbations of the Moore-Penrose metric generalized inverses of linear operators in Banach spaces are described. The Moore Penrose metric generalized inverse is homo- geneous and nonlinear in general, and the proofs of our results are different from linear generalized inverses. By using the quasi-additivity of Moore-Penrose metric generalized inverse and the theorem of generalized orthogonal decomposition, we show some error estimates of perturbations for the single- valued Moore-Penrose metric generalized inverses of bounded linear operators. Furthermore, by means of the continuity of the metric projection operator and the quasi-additivity of Moore-Penrose metric generalized inverse, an expression for Moore-Penrose metric generalized inverse is given.

Perturbation Analysis of Moore–Penrose Quasi-linear Projection Generalized Inverse of Closed Linear Operators in Banach Spaces

In this paper, we investigate the perturbation problem for the Moore-Penrose bounded quasi-linear projection generalized inverses of a closed linear operaters in Banach space. By the method of the perturbation analysis of bounded quasi-linear operators, we obtain an explicit perturbation theorem and error estimates for the Moore-Penrose bounded quasi-linear generalized inverse of closed linear operator under the T-bounded perturbation, which not only extend some known results on the perturbation of the oblique projection generalized inverse of closed linear operators, but also extend some known results on the perturbation of the Moore-Penrose metric generalized inverse of bounded linear operators in Banach spaces.

The Generalized Regular Points and Narrow Spectrum Points of Bounded Linear Operators on Hilbert Spaces

In this paper, we introduce the concepts of generalized regular points and narrow spectrum points of bounded linear operators on Hilbert spaces. The concept of generalized regular points is an extension of the concept regular points, and so, the set of all spectrum points is reduced to the narrow spectrum. We present not only the same and different properties of spectrum and of narrow spectrum but also show the relationship between them. Finally, the well known problem about the invariant subspaces of bounded linear operators on separable Hilbert spaces is simplified to the problem of the operator with narrow spectrum only.

Existence and Uniqueness of Positive Solutions for a Class of Semilinear Elliptic Systems

The authors prove the uniqueness and existence of positive solutions for the semilinear elliptic system which involves nonlinearities with sublinear growth conditions.

Criteria for the Single-Valued Metric Generalized Inverses of Multi-Valued Linear Operators in Banach Spaces

Let X, Y be Banach spaces and M be a linear subspace in X x Y = {{x,y}lx E X,y C Y}. We may view M as a multi-valued linear operator from X to Y by taking M（x） = {yl（x,y} C M}. In this paper, we give several criteria for a single-valued operator from Y to X to be the metric generalized inverse of the multi-valued linear operator M. The principal tool in this paper is also the generalized orthogonal decomposition theorem in Banach spaces.