The best possibility of the grand Furuta inequality期刊界 All Journals 搜尽天下杂志传播学术成果专业期刊搜索期刊信息化学术搜索

The best possibility of the grand Furuta inequality

Authors:	Kô tarô Tanahashi

Affiliation:	Department of Mathematics, Tohoku College of Pharmacy, Komatsushima, Aoba-ku, Sendai 981-8558, Japan

Abstract:	Let be invertible bounded linear operators on a Hilbert space satisfying , and let be real numbers satisfying Furuta showed that if $0 < alpha leq dfrac{ 1-t+r}{ (p-t)s + r}$ , then $left{ A^{frac{r}{2}} left( A^{ -frac{t}{2}} B^{p} A^{ -frac{t}{2}} right)^{s} A^{frac{r}{2}} right}^{alpha } leq A^{ left{ (p-t)s + r right} alpha }$ . This inequality is called the grand Furuta inequality, which interpolates the Furuta inequality and the Ando-Hiai inequality ( ). In this paper, we show the grand Furuta inequality is best possible in the following sense: that is, if $dfrac{ 1-t+r}{ (p-t)s + r} < alpha$ , then there exist invertible matrices with which do not satisfy $left{ A^{frac{r}{2}} left( A^{ -frac{t}{2}} B^{p} A^{ -frac{t}{2}} right)^{s} A^{frac{r}{2}} right}^{alpha } leq A^{ left{ (p-t)s + r right} alpha }$ .

Keywords:	The L" owner-Heinz inequality, the Furuta inequality, the grand Furuta inequality

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