The operator equation

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 ; (ii) a necessary and sufficient condition for the existence of such *T* is that for some , and then ; (iii) this condition is satisfied if *H* is invertible or more generally if for some ; (iv) an exact formula for *T* is given when *H* is invertible.

**[1]**Gert K. Pedersen,*Some operator monotone functions*, Proc. Amer. Math. Soc.**36**(1972), 309–310. MR**0306957**, 10.1090/S0002-9939-1972-0306957-4**[2]**Gert K. Pedersen and Masamichi Takesaki,*The Radon-Nikodym theorem for von Neumann algebras*, Acta Math.**130**(1973), 53–87. MR**0412827****[3]**Shôichirô Sakai,*A Radon-Nikodým theorem in 𝑊*-algebras*, Bull. Amer. Math. Soc.**71**(1965), 149–151. MR**0174992**, 10.1090/S0002-9904-1965-11265-4**[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**

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DOI:
https://doi.org/10.1090/S0002-9939-1972-0306958-6

Keywords:
Noncommutative Radon-Nikodym theorem,
positive operators

Article copyright:
© Copyright 1972
American Mathematical Society