A new result on reverse order laws for {1,2,3}-inverse of a two-operator product
Abstract
In this note, reverse order laws for $\{1,2,3\}$-inverse of a two-operator product is mainly investigated by making full use of block-operator matrix technique. First, an example is given, which demonstrates there is a gap in the main result in [{\it {X. J. Liu, S. X. Wu, D. S. Cvetkovi$\acute{c}$-Ili$\acute{c}$. New results on reverse order law for $\{1,2,3\}$- and $\{1,2,4\}$-inverses of bounded operators. Mathematics of Computation, 2013, 82(283): 1597-1607}}]. Next, The new necessary and sufficient conditions for $B\{1,2,i\}A\{1,2,i\}\subseteq(AB)\{1,2,i\}(i\in\{3,4\})$ are presented respectively, when all ranges $R(A), \ R(B)$ and $ R(AB)$ are closed. Which will fill up the gap in the above paper.
Copyright © 2017 Haiyan Zhang, Yuejuan Sun, Weiyan Yu. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Cite this article
H. Zhang, Y. Sun, W. Yu, A new result on reverse order laws for {1,2,3}-inverse of a two-operator product, J. Math. Comput. Sci. 7 (2017), 1006–1021.
BibTeX
@article{Zhang2017,
author = {Zhang, Haiyan and Sun, Yuejuan and Yu, Weiyan},
title = {A new result on reverse order laws for {1,2,3}-inverse of a two-operator product},
journal = {J. Math. Comput. Sci.},
volume = {7},
year = {2017},
pages = {1006--1021},
doi = {}
}