A decreasing operator function associated with the Furuta inequality
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- by Takayuki Furuta and Derming Wang PDF
- Proc. Amer. Math. Soc. 126 (1998), 2427-2432 Request permission
Abstract:
Let $A\ge B\ge 0$ with $A>0$ and let $t\in [0,1]$ and $q\ge 0$. As a generalization of a result due to Furuta, it is shown that the operator function \[ G_{p,q,t}(A,B,r,s)=A^{-r/2}\{A^{r/2} (A^{-t/2} B^pA^{-t/2})^s A^{r/2}\}^{(q-t+r)/[(p-t)s+r]}A^{-r/2} \] is decreasing for $r\ge t$ and $s\ge 1$ if $p\ge \max \{q,t\}$. Moreover, if $1\ge p>t$ and $q\ge t$, then $G_{p,q,t}(A,B,r,s)$ is decreasing for $r\ge 0$ and $s\ge \frac {q-t}{p-t}$. The latter result is an extension of an earlier result of Furuta.References
- Tsuyoshi Ando and Fumio Hiai, Log majorization and complementary Golden-Thompson type inequalities, Linear Algebra Appl. 197/198 (1994), 113โ131. Second Conference of the International Linear Algebra Society (ILAS) (Lisbon, 1992). MR 1275611, DOI 10.1016/0024-3795(94)90484-7
- Masatoshi Fujii, Furutaโs inequality and its mean theoretic approach, J. Operator Theory 23 (1990), no.ย 1, 67โ72. MR 1054816
- Masatoshi Fujii, Takayuki Furuta, and Eizaburo Kamei, Furutaโs inequality and its application to Andoโs theorem, Linear Algebra Appl. 179 (1993), 161โ169. MR 1200149, DOI 10.1016/0024-3795(93)90327-K
- Masatoshi Fujii and Eizaburo Kamei, Mean-theoretic approach to the grand Furuta inequality, Proc. Amer. Math. Soc. 124 (1996), no.ย 9, 2751โ2756. MR 1327013, DOI 10.1090/S0002-9939-96-03342-4
- 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.ย 1, 85โ88. MR 897075, DOI 10.1090/S0002-9939-1987-0897075-6
- Takayuki Furuta, A proof via operator means of an order preserving inequality, Linear Algebra Appl. 113 (1989), 129โ130. MR 978587, DOI 10.1016/0024-3795(89)90291-7
- Takayuki Furuta, An elementary proof of an order preserving inequality, Proc. Japan Acad. Ser. A Math. Sci. 65 (1989), no.ย 5, 126. MR 1011850
- Takayuki Furuta, Two operator functions with monotone property, Proc. Amer. Math. Soc. 111 (1991), no.ย 2, 511โ516. MR 1045135, DOI 10.1090/S0002-9939-1991-1045135-2
- Takayuki Furuta, Applications of order preserving operator inequalities, Operator theory and complex analysis (Sapporo, 1991) Oper. Theory Adv. Appl., vol. 59, Birkhรคuser, Basel, 1992, pp.ย 180โ190. MR 1246815
- Takayuki Furuta, Extension of the Furuta inequality and Ando-Hiai log-majorization, Linear Algebra Appl. 219 (1995), 139โ155. MR 1327396, DOI 10.1016/0024-3795(93)00203-C
- Takayuki Furuta, Parallelism related to the inequality โ$A\geq B\geq 0$ ensures $(A^{r/2}A^pA^{r/2})^{(1+r)/(p+r)}\geq (A^{r/2}B^pA^{r/2})^{(1+r)/(p+r)}$ for $p\geq 1$ and $r\geq 0$โ, Math. Japon. 45 (1997), no.ย 2, 203โ209. MR 1441493
- Frank Hansen, An operator inequality, Math. Ann. 246 (1979/80), no.ย 3, 249โ250. MR 563403, DOI 10.1007/BF01371046
- C. J. Everett Jr., Annihilator ideals and representation iteration for abstract rings, Duke Math. J. 5 (1939), 623โ627. MR 13
- Eizaburo Kamei, A satellite to Furutaโs inequality, Math. Japon. 33 (1988), no.ย 6, 883โ886. MR 975867
- K. Lรถwner, รber monotone Matrixfunktionen, Math. Z. 38 (1934), 177โ216.
- Kรดtarรด Tanahashi, Best possibility of the Furuta inequality, Proc. Amer. Math. Soc. 124 (1996), no.ย 1, 141โ146. MR 1291794, DOI 10.1090/S0002-9939-96-03055-9
Additional Information
- Takayuki Furuta
- Affiliation: Department of Applied Mathematics, Faculty of Science, Science University of Tokyo, Kagurazaka, Shinjuku 162-8601, Tokyo, Japan
- Email: furuta@rs.kagu.sut.ac.jp
- Derming Wang
- Affiliation: Department of Mathematics, California State University, Long Beach, Long Beach, California 90840-1001
- Received by editor(s): January 23, 1997
- Communicated by: Palle E. T. Jorgensen
- © Copyright 1998 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 126 (1998), 2427-2432
- MSC (1991): Primary 47A63
- DOI: https://doi.org/10.1090/S0002-9939-98-04632-2
- MathSciNet review: 1473667