Linear Algebra and its Applications, ( ISI ), Volume (517), No (1), Year (2017-1) , Pages (85-98)

Title : ( Operator equations AX+YB=C and AXA^*+BYB^*=C in Hilbert C^*-modules )

Authors: Z. Mousavi , R. Eskandari , Mohammad Sal Moslehian , F. Mirzapour ,

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Abstract

Let $A,B$ and $C$ be adjointable operators on a Hilbert $C^*$-module $\mathscr{E}$. Giving a suitable version of the celebrated Douglas theorem in the context of Hilbert $C^*$-modules, we present the general solution of the equation $AX+YB=C$ when the ranges of $A,B$ and $C$ are not necessarily closed. We examine a result of Fillmore and Williams in the setting of Hilbert $C^*$-modules. Moreover, we obtain some necessary and sufficient conditions for existence of a solution for $AXA^*+BYB^*=C$. Finally, we deduce that there exist nonzero operators $X, Y\geq 0$ and $Z$ such that $AXA^*+BYB^*=CZ$, when $A, B$ and $C$ are given subject to some conditions.

Keywords

, Hilbert $C^*$, module; Operator equation; Solution; Orthogonally complemented
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@article{paperid:1060681,
author = {Z. Mousavi and R. Eskandari and Sal Moslehian, Mohammad and F. Mirzapour},
title = {Operator equations AX+YB=C and AXA^*+BYB^*=C in Hilbert C^*-modules},
journal = {Linear Algebra and its Applications},
year = {2017},
volume = {517},
number = {1},
month = {January},
issn = {0024-3795},
pages = {85--98},
numpages = {13},
keywords = {Hilbert $C^*$-module; Operator equation; Solution; Orthogonally complemented},
}

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%0 Journal Article
%T Operator equations AX+YB=C and AXA^*+BYB^*=C in Hilbert C^*-modules
%A Z. Mousavi
%A R. Eskandari
%A Sal Moslehian, Mohammad
%A F. Mirzapour
%J Linear Algebra and its Applications
%@ 0024-3795
%D 2017

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