Γ-inverses of bounded linear operators

Let be the algebra of all the bounded linear operators on a Hilbert space . For A , P and Q in , if there exists an operator such that APXQA = A , XQAPX = X , ( QAPX )* = QAPX and ( XQAP )* = XQAP , then X is said to be the Γ-inverse of A associated with P and Q , and denoted by A P , Q + . In this...

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Bibliographic Details
Published in:Acta mathematica Sinica. English series 2014-04, Vol.30 (4), p.675-680
Main Authors: Xu, Xiao Ming, Du, Hong Ke, Fang, Xiao Chun
Format: Article
Language:English
Subjects:
Algebra
Hilbert space
Linear operators
Mathematics
Mathematics and Statistics
Operators
Representations
Texts
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Summary:Let be the algebra of all the bounded linear operators on a Hilbert space . For A , P and Q in , if there exists an operator such that APXQA = A , XQAPX = X , ( QAPX )* = QAPX and ( XQAP )* = XQAP , then X is said to be the Γ-inverse of A associated with P and Q , and denoted by A P , Q + . In this note, we present some necessary and sufficient conditions for which A P , Q + exists, and give an explicit representation of A P , Q + (if A P , Q + exists).
ISSN:1439-8516
1439-7617
DOI:10.1007/s10114-013-2552-y