Proceedings of the American Mathematical Society, ( ISI ), Volume (148), No (3), Year (2019-1) , Pages (1139-1151)

Title : ( Douglas factorization theorem revisited )

Authors: Vladimir Manuilov , Mohammad Sal Moslehian , Qingxiang Xu ,

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Abstract

Inspired by the Douglas factorization theorem, we investigate the solvability of the operator equation AX = C in the framework of Hilbert C- modules. Utilizing partial isometries, we present its general solution when A is a semi-regular operator. For such an operator A, we show that the equation AX = C has a positive solution if and only if the range inclusion R(C) R(A) holds and CC tCA for some t > 0. In addition, we deal with the solvability of the operator equation (P + Q)1=2X = P, where P and Q are projections. We provide a tricky counterexample to show that there exist a C-algebra A, a Hilbert A-module H and projections P and Q on H such that the operator equation (P + Q)1=2X = P has no solution. Moreover, we give a perturbation result related to the latter equation.

Keywords

, Hilbert C-module, operator equation, regular operator, semi-regular operator
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@article{paperid:1076442,
author = {Vladimir Manuilov and Sal Moslehian, Mohammad and Qingxiang Xu},
title = {Douglas factorization theorem revisited},
journal = { Proceedings of the American Mathematical Society},
year = {2019},
volume = {148},
number = {3},
month = {January},
issn = {0002-9939},
pages = {1139--1151},
numpages = {12},
keywords = {Hilbert C-module; operator equation; regular operator; semi-regular operator},
}

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%0 Journal Article
%T Douglas factorization theorem revisited
%A Vladimir Manuilov
%A Sal Moslehian, Mohammad
%A Qingxiang Xu
%J Proceedings of the American Mathematical Society
%@ 0002-9939
%D 2019

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