Establishment of operator inequalities by using computers and their applications

利用计算机及其应用建立算子不等式

基本信息

批准号：
11640186
负责人：
FURUTA Takayuki
金额：
$ 2.3万
依托单位：
Science University of Tokyo
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (C)
财政年份：
1999
资助国家：
日本
起止时间：
1999 至 2000
项目状态：
已结题

来源：
https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-11640186/
关键词：
Lowner-Heinz inequality Furuta inequality generalized Furuta inequality log majorization order preserving inequality chaotic order relative operator entropy positive definite operator Lowner-Heinz不等式 relative entropy log-hyponormal p

项目摘要

In what follows, a capital letter means a bounded linear operator on a Hilbert space. Furuta inequality (1987) asserts that if A 【greater than or equal】 B 【greater than or equal】 0, then for r 【greater than or equal】 0,(*) (A^<r/2> A^p A^<r/2>)^<1/q> 【greater than or equal】 (A^<r/2> B^p A^<r/2>)^<1/q>holds for p 【greater than or equal】 0 and q 【greater than or equal】 1 with (1 + r)q 【greater than or equal】 p + r. Furuta inequality yields the famous Lowner-Heinz one (1934), that is, A 【greater than or equal】 B 【greater than or equal】 0 ensures A^p 【greater than or equal】 B^p for 1 【greater than or equal】 p 【greater than or equal】 0 when we put r = 0 in (*). We obtained a lot of applications of Furuta inequality in the following three branches, (a) operator ibnequalities, (b) norm inequalities and (c) operator equations. We cite some of them as follows : (a_1) relative operator entropy, (a_2) Ando-Hiai log majorization, (a_3) Aluthge transformation, (b_1) Heinz-Kato inequality, (b_2) Kosaki trace inequality, (c_1) Pedersen-Takesaki operator equation. Recently we obtained a one page simplified proof of generalized Furuta inequality which interpolates Furuta inequality itself and an inequality equivalent to the main theorem on log majorizaton by Ando-Hiai. Further applications of Furuta inequality to some operator equatios and relative operator entoropy will be expected in near future.

在接下来的内容中，大写字母意味着希尔伯特空间上的有界线性操作员。 Furuta不平等（1987）断言，如果A [大于或相等] B [大于或等于] 0，则对于R [大于或等于或等于] 0，（*）（a^<r/2> a^p a^p a^p a^<r/2>）^<1/q>> [大于或等于或等于]（大于或均等）与（1 + r）q [大于或相等] p + r等于] 1。 Furuta不平等产生了著名的下海因兹ONE（1934），即a [大于或相等] b [大于或相等] 0确保a^p [大于或相等] b^p对于1 [大于或等于] p [大于或等于] p [大于或等于] 0时，当我们将r = 0放在（*中）时。我们在以下三个分支中获得了Furuta不等式的许多应用，（a）操作员ibnequalities，（b）规范不等式和（c）操作员方程。我们引用了其中的一些如下：（A_1）相对操作员熵，（A_2）Ando-Hiai Log矩阵，（A_3）Aluthge Transformation，（B_1）Heinz-Kato不平等，（B_2）Kosaki Trace Intorality不平等，（C_1）PEDERSEN-TAKESEN-TAKESEN-TAKESEN-TAKESEN-TAKESEN-TAKESEN-TAKESEAKI OPERATORETORATORE EQUATIONETORATIE。最近，我们获得了一页简化的普遍不平等的证明，该证明将Furuta的不平等本身介绍，而不平等现象与Ando-Hiai对log Majorizaton上的主要定理相当。 Furuta不平等的进一步应用在某些运营商等电机和相对操作员内的范围内预计将在不久的将来。