The operator equation $THT=K$
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- by Gert K. Pedersen and Masamichi Takesaki PDF
- Proc. Amer. Math. Soc. 36 (1972), 311-312 Request permission
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.References
- Gert K. Pedersen, Some operator monotone functions, Proc. Amer. Math. Soc. 36 (1972), 309–310. MR 306957, DOI 10.1090/S0002-9939-1972-0306957-4
- Gert K. Pedersen and Masamichi Takesaki, The Radon-Nikodym theorem for von Neumann algebras, Acta Math. 130 (1973), 53–87. MR 412827, DOI 10.1007/BF02392262
- Shôichirô Sakai, A Radon-Nikodým theorem in $W^{\ast }$-algebras, Bull. Amer. Math. Soc. 71 (1965), 149–151. MR 174992, DOI 10.1090/S0002-9904-1965-11265-4
- M. Takesaki, Tomita’s theory of modular Hilbert algebras and its applications, Lecture Notes in Mathematics, Vol. 128, Springer-Verlag, Berlin-New York, 1970. MR 0270168
Additional Information
- © Copyright 1972 American Mathematical Society
- 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