Mixed-Type Reverse Order Laws Associated to $\{1,3,4\}$-Inverse
Received:December 16, 2018  Revised:May 26, 2019
Key Word: $\{1,3,4\}$-inverse   reverse order law   generalized inverse   block-operator matrix  
Fund ProjectL:Supported by the National Natural Science Foundation of China (Grant Nos.11501345; 11671261) and the Youth Backbone Teacher Training Program of Henan Province (Grant No.2017GGJS140).
Author NameAffiliation
Haiyan ZHANG School of Mathematics and Statistics, Shangqiu Normal University, Henan 476000, P. R. China 
Chunyuan DENG School of Mathematics Sciences, South China Normal University, Guangdong 510631, P. R. China 
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      In this paper, we study the mixed-type reverse order laws to $\{1,3,4\}$-inverses for closed range operators $A$, $B$ and $AB$. It is shown that $B\{1,3,4\}A\{1,3,4\}\subseteq (AB)\{1,3\}$ if and only if $R(A^*AB)\subseteq R(B)$. For every $A^{(134)}\in A\{1,3,4\}$, it has $(A^{(134)}AB)\{1,3,4\}A\{1,3,4\}= (AB)\{1,3,4\}$ if and only if $R(AA^*AB)\subseteq R(AB)$. As an application of our results, some new characterizations of the mixed-type reverse order laws associated to the Moore-Penrose inverse and the $\{1,3,4\}$-inverse are established.
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