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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
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\Vert T \right\Vert \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.

References [Enhancements On Off] (What's this?)

  • [1] G. K. Pedersen, Some operator monotone functions, Proc. Amer. Math. Soc. 36 (1972), 309-310. MR 0306957 (46:6078)
  • [2] G. K. Pedersen and M. Takesaki, The Radon-Nikodym theorem for von Neumann algebras, Acta Math. (to appear). MR 0412827 (54:948)
  • [3] S. Sakai, A Radon-Nikodym theorem in $ {W^ \ast }$-algebras, Bull. Amer. Math. Soc. 71 (1965), 149-151. MR 30 #5180. MR 0174992 (30:5180)
  • [4] M. Takesaki, Tomita's theory of modular Hilbert algebras and its applications, Lecture Notes in Math., vol. 128, Springer-Verlag, Berlin and New York, 1970. MR 42 #5061. MR 0270168 (42:5061)

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

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