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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

The operator equation $THT=K$


Authors: Gert K. Pedersen and Masamichi Takesaki
Journal: Proc. Amer. Math. Soc. 36 (1972), 311-312
MSC: Primary 47B15; Secondary 47A65
DOI: https://doi.org/10.1090/S0002-9939-1972-0306958-6
MathSciNet review: 0306958
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Abstract: Let H and K be bounded positive operators on a Hilbert space, and assume that H is nonsingular. Then (i) there is at most one bounded positive operator T such that $THT = K$; (ii) a necessary and sufficient condition for the existence of such T is that ${({H^{1/2}}K{H^{1/2}})^{1/2}} \leqq aH$ for some $a > 0$, and then $\left \| T \right \| \leqq a$; (iii) this condition is satisfied if H is invertible or more generally if $K \leqq {a^2}H$ for some $a > 0$; (iv) an exact formula for T is given when H is invertible.


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Keywords: Noncommutative Radon-Nikodym theorem, positive operators
Article copyright: © Copyright 1972 American Mathematical Society