Best possibility of the Furuta inequality
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- by Kôtarô Tanahashi
- Proc. Amer. Math. Soc. 124 (1996), 141-146
- DOI: https://doi.org/10.1090/S0002-9939-96-03055-9
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Abstract:
Let $0\le p,q,r\in \Bbb R, p+2r\le (1+2r)q$, and $1\le q$. Furuta (1987) proved that if bounded linear operators $A,B\in B(H)$ on a Hilbert space $H$ $(\dim (H)\ge 2)$ satisfy $0\le B\le A$, then $(A^r B^p A^r)^{1/q} \le A^{(p+2r)/q}$. In this paper, we prove that the range $p+2r\le (1+2r)q$ and $1\le q$ is best possible with respect to the Furuta inequality, that is, if $(1+2r) q<p+2r$ or $0<q<1$, then there exist $A,B\in B(\Bbb R^2)$ which satisfy $0\le B\le A$ but $(A^r B^p A^r)^{1/q}\nleq A^{(p+2r)/q}$.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
- C. J. Everett Jr., Annihilator ideals and representation iteration for abstract rings, Duke Math. J. 5 (1939), 623–627. MR 13
- K. Löwner, Über monotone Matrixfunktionen, Math. Z. 38 (1934), 177–216.
Bibliographic Information
- Kôtarô Tanahashi
- Affiliation: Department of Mathematics, Tohoku College of Pharmacy, Komatsushima, Aoba-ku, Sendai 981, Japan
- Received by editor(s): February 25, 1994
- Received by editor(s) in revised form: July 7, 1994
- Communicated by: Palle E. T. Jorgensen
- © Copyright 1996 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 124 (1996), 141-146
- MSC (1991): Primary 47B15
- DOI: https://doi.org/10.1090/S0002-9939-96-03055-9
- MathSciNet review: 1291794