闭值域算子的Moore-Penrose逆的谱分解

Spectral decomposition of the Moore-Penrose inverse of operator with closed range

摘要

首先证明算子Moore-Penrose逆的几种定义之间的等价性。其次利用自伴算子的谱分解,给出闭值域算子Moore-Penrose逆的谱分解。最后利用算子Moore-Penrose逆给出最佳逼近集的一般表示,证明最佳逼近集是一个仿射流形。

Abstract

First, this paper proves the equivalence among several definitions of the Moore-Penrose inverse of an operator. Second, by utilizing the spectral decomposition of a self-adjoint operator, we provide the spectral decomposition of the Moore-Penrose inverse of an operator with a closed range. Finally, by employing the Moore-Penrose inverse of an operator, we offer a general representation of the best approximation set, thereby proving that the best approximation set is an affine manifold.

关键词

算子Moore-Penrose逆 / 谱分解 / 仿射流形

Key words

Moore-Penrose inverse of operator / spectral decomposition / affine manifolds

参考文献

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基金资助

陕西省自然科学基金项目(2023-JC-YB-627)

国家自然科学基金资助项目(12061031)