Equations ax = c and xb = d in rings and rings with involution with applications to Hilbert space operators

被引：42

作者：

Dajic, Alegra ^{[1
]}

Koliha, J. J. ^{[1
]}

机构：

[1] Univ Melbourne, Dept Math & Stat, Melbourne, Vic 3010, Australia

来源：

LINEAR ALGEBRA AND ITS APPLICATIONS | 2008年 / 429卷 / 07期

关键词：

ring; ring with involution; equations in a ring; Hermitian solution; reflexive solution; matrix equations; operator equations; Hilbert space operators; positive solution; real-positive solution;

D O I：

10.1016/j.laa.2008.05.012

中图分类号：

O29 [应用数学];

学科分类号：

070104 ;

摘要：

This paper reviews the equations ax = c and xb = d from a new perspective by studying them in the setting of associative rings with or without involution. Results for rectangular matrices and operators between different Banach and Hilbert spaces are obtained by embedding the 'rectangles' into rings of square matrices or rings of operators acting on the same space. Necessary and sufficient conditions using generalized inverses are given for the existence of the Hermitian, skew-Hermitian, reflexive, antireflexive, positive and real-positive solutions, and the general solutions are described in terms of the original elements or operators. New results are obtained, and many results existing in the literature are recovered and corrected. (c) 2008 Elsevier Inc. All rights reserved.

引用

页码：1779 / 1809

页数：31

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