Mixed-type reverse order laws associated to {1,3,4}-inverse
Received:December 16, 2018  Revised:May 05, 2019
Key Word: {1,3,4}-inverse   reverse order law   generalized inverse   block-operator matrix.
Fund ProjectL:The National Natural Science Foundation of China （No.11501345, No.11671261）
 Author Name Affiliation E-mail Haiyan Zhang School of Mathematics and Statistics, Shangqiu Normal University csqam@163.com Chunyuan Deng College of Mathematics Sciences, South China Normal University
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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 one 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.