Electronic Journal of Linear Algebra, ( ISI ), Volume (32), No (1), Year (2017-5) , Pages (172-183)

Title : ( Solutions of the system of operator equations BXA=B=AXB via *-order )

Authors: mehdi vosough , Mohammad Sal Moslehian ,

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Abstract

In this paper, some necessary and sufficient conditions are established for the existence of solutions to the system of operator equations $BXA=B=AXB$ in the setting of bounded linear operators on a Hilbert space, where the unknown operator $X$ is called the inverse of $A$ along $B$. After that, under some mild conditions, it is proved that an operator $X$ is a solution of $BXA=B=AXB$ if and only if $B \stackrel{*}{ \leq} AXA$, where the $*$-order $C\stackrel{*}{ \leq} D$ means $CC^*=DC^*, C^*C=C^*D$. Moreover, the general solution of the equation above is obtained. Finally, some characterizations of $C \stackrel{*}{ \leq} D$ via other operator equations, are presented.

Keywords

, $*$-Order, Moore--Penrose inverse, Matrix equation, Operator equation
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@article{paperid:1064445,
author = {Vosough, Mehdi and Sal Moslehian, Mohammad},
title = {Solutions of the system of operator equations BXA=B=AXB via *-order},
journal = {Electronic Journal of Linear Algebra},
year = {2017},
volume = {32},
number = {1},
month = {May},
issn = {1537-9582},
pages = {172--183},
numpages = {11},
keywords = {$*$-Order; Moore--Penrose inverse; Matrix equation; Operator equation},
}

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%0 Journal Article
%T Solutions of the system of operator equations BXA=B=AXB via *-order
%A Vosough, Mehdi
%A Sal Moslehian, Mohammad
%J Electronic Journal of Linear Algebra
%@ 1537-9582
%D 2017

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