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Douglas factorization theorem revisited


Authors: Vladimir Manuilov, M. S. Moslehian and Qingxiang Xu
Journal: Proc. Amer. Math. Soc. 148 (2020), 1139-1151
MSC (2010): Primary 47A62; Secondary 46L08, 47A05
DOI: https://doi.org/10.1090/proc/14757
Published electronically: September 20, 2019
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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 $ {\mathcal R}(C) \subseteq {\mathcal R}(A)$ holds and $ CC^*\le t\, CA^*$ for some $ t>0$. In addition, we deal with the solvability of the operator equation $ (P+Q)^{1/2}X=P$, where $ P$ and $ Q$ are projections. We provide a tricky counterexample to show that there exist a $ C^*$-algebra $ \mathfrak{A}$, a Hilbert $ \mathfrak{A}$-module $ \mathscr {H}$, and projections $ P$ and $ Q$ on $ \mathscr {H}$ such that the operator equation $ (P+Q)^{1/2}X=P$ has no solution. Moreover, we give a perturbation result related to the latter equation.


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Additional Information

Vladimir Manuilov
Affiliation: Department of Mechanics and Mathematics, Moscow State University, Moscow, 119991, Russia
Email: manuilov@mech.math.msu.su

M. S. Moslehian
Affiliation: Department of Pure Mathematics, Ferdowsi University of Mashhad, Center of Excellence in Analysis on Algebraic Structures (CEAAS), P.O. Box 1159, Mashhad 91775, Iran
Email: moslehian@um.ac.ir

Qingxiang Xu
Affiliation: Department of Mathematics, Shanghai Normal University, Shanghai 200234, People’s Republic of China
Email: qxxu@shnu.edu.cn; qingxiang_xu@126.com

DOI: https://doi.org/10.1090/proc/14757
Keywords: Hilbert $C^*$-module, operator equation, regular operator, semi-regular operator
Received by editor(s): January 31, 2019
Received by editor(s) in revised form: July 1, 2019
Published electronically: September 20, 2019
Additional Notes: The first author was partially supported by the RFBR grant No. 19-01-00574.
The second author was partially supported by a grant from Ferdowsi University of Mashhad (No. 2/50300).
The third author was partially supported by a grant from Shanghai Municipal Science and Technology Commission (18590745200).
Dedicated: Dedicated to the memory of R. G. Douglas (1938-2018)
Communicated by: Stephan Ramon Garcia
Article copyright: © Copyright 2019 American Mathematical Society