Common Hermitian solutions to some operator equations on Hilber C ∗ -modules

We establish necessary and sufficient conditions for the existence of the general common Hermitian solution to the equations A 1 X = C 1 , XB 1 = C 2 , A 3 XA 3 ∗ = C 3 , A 4 XA 4 ∗ = C 4 for adjointable operators between Hilbert C ∗ -modules, and present an expression for the common Hermitian solut...

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Bibliographic Details
Published in:Linear algebra and its applications 2010-07, Vol.432 (12), p.3159-3171
Main Authors: Wang, Qing-Wen, Wu, Zhong-Cheng
Format: Article
Language:English
Subjects:
Hermitian solution
Hilbert [formula omitted]-module
Moore–Penrose inverse
Operator equation
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Summary:We establish necessary and sufficient conditions for the existence of the general common Hermitian solution to the equations A 1 X = C 1 , XB 1 = C 2 , A 3 XA 3 ∗ = C 3 , A 4 XA 4 ∗ = C 4 for adjointable operators between Hilbert C ∗ -modules, and present an expression for the common Hermitian solution to the equations in terms of Moore–Penrose inverse of operators when the solvability conditions are satisfied. The findings of this paper extend some known results in the literature.
ISSN:0024-3795
DOI:10.1016/j.laa.2010.01.015