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Two operator functions with monotone property


Author: Takayuki Furuta
Journal: Proc. Amer. Math. Soc. 111 (1991), 511-516
MSC: Primary 47A60; Secondary 47B15
DOI: https://doi.org/10.1090/S0002-9939-1991-1045135-2
MathSciNet review: 1045135
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Abstract: This paper proves that $ F(p) = {({B^r}{A^p}{B^r})^{(1 + 2r)/(p + 2r)}}$ is an increasing function of $ p$ for $ p \geqq 1$ and $ r \geqq 0$ whenever $ A \geqq B \geqq 0$. This result is more precise than our previous result that $ A \geqq B \geqq 0$ ensures $ {({B^r}{A^p}{B^r})^{(1 + 2r)/(p + 2r)}} \geqq {B^{1 + 2r}}$ for each $ p \geqq 1$ and $ r \geqq 0$. We also cite three counterexamples related to Theorem 1.


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

  • [1] T. Furuta, $ A \geqq B \geqq 0$ assures $ {({B^r}{A^p}{B^r})^{1/q}} \geqq {B^{(p + 2r)/q}}$ for $ r \geqq 0,p \geqq 0,q \geqq 1$ with $ (1 + 2r)q \geqq p + 2r$, Proc. Amer. Math. Soc. 101 (1987), 85-88. MR 897075 (89b:47028)
  • [2] K. Löwner, Über monotone Matrixfunktionen, Math. Z. 38 (1934), 177-216. MR 1545446
  • [3] G. K. Pedersen, Some operator monotone functions, Proc. Amer. Math. Soc. 36 (1972), 309-310. MR 0306957 (46:6078)

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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1991-1045135-2
Keywords: Positive operator, order-preserving inequalities
Article copyright: © Copyright 1991 American Mathematical Society

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