Banach空间中线性算子的(集值)度量广义逆及其齐性单值选择

The Set-Valued Metric Generalized Inverse of Linear Operator in Banach Space and Its Bounded Homogeneous Single Selection

为研究Banach空间中不适定线性算子方程的最佳逼近解,Nashed在文[1]中引入了Banach空问中线性算子T的(集值)度量广义逆T的概念,并提出“求解线性算子的(集值)度量广义逆的具有良好性质的单值选择是值得研究”的公开问题.本文首先证明了Banach空间中线性算子的度量广义逆是具有闭凸值的集值映射,给出了该度量广义逆的等价表达式,并利用Banach空间的再赋范方法,给出其有界齐性的单值选择,部分地解决了Nashed所提出的公开问题.

In this paper, the representation of the set-valued metric generalized inverse of closed linear operator with closed range in reflexive Banach space and its single selection are given. By the geometric property of the space, we prove that single selection of set-valued metric generalized inverse is a bounded homogeneous operator.