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.

https://doi.org/10.28919/jmcs/3483