Two operator functions with monotone property
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- by Takayuki Furuta
- Proc. Amer. Math. Soc. 111 (1991), 511-516
- DOI: https://doi.org/10.1090/S0002-9939-1991-1045135-2
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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
- Takayuki Furuta, $A\geq B\geq 0$ assures $(B^rA^pB^r)^{1/q}\geq B^{(p+2r)/q}$ for $r\geq 0$, $p\geq 0$, $q\geq 1$ with $(1+2r)q\geq p+2r$, Proc. Amer. Math. Soc. 101 (1987), no. 1, 85–88. MR 897075, DOI 10.1090/S0002-9939-1987-0897075-6
- Karl Löwner, Über monotone Matrixfunktionen, Math. Z. 38 (1934), no. 1, 177–216 (German). MR 1545446, DOI 10.1007/BF01170633
- 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
Bibliographic Information
- © Copyright 1991 American Mathematical Society
- 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