Establishment of operator inequalities on Hilbert space and their applications

希尔伯特空间算子不等式的建立及其应用

• 批准号: 09640225
• 负责人: FURUTA Takayuki
• 金额: $ 1.98万
• 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: 1997
• 资助国家: 日本
• 起止时间: 1997 至 1998
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-09640225/
• 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 inequality; Lowner-Heinz inequaity; Rela; Lowner-Heinz inequality; Furuta inequality; generalized Furuta inequality; log majorization; order preserving inequality; chaotic order; relative operator entropy; positive definite operator; Lowner-Heinz inequality; Lowner-Heinz inequaity; Rela

In what follows, a capital letter means a bounded linear operator on a Hulbert space. Furuta inequality (1987) asserts that if A <greater than or equal> B <greater than or equal> 0, then for r(*)(A^<r/>A^pA^<r/>)^<1/> <greater than or equal> (A^<r/>B^pA^<r/>)^<1/>holds for p <greater than or equal> 0 and q <greater than or equal> 1 with (l+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 : (alpha_1) relative operator entropy, (alpha_2) Ando-Hiai log majorization, (alpha_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 and equivalence relations among this generalized Furuta inequality and related operator functions. We have to consider some relations among Kantrovich inequality, Holder-McCarthy inequality, Reid inequality, Jensen inequality and Furuta inequality. Further applications and developments of Furuta inequality will be expected

在接下来的内容中，大写字母是指赫尔伯特空间上有界的线性操作员。 Furuta不等式（1987）断言，如果a <大于或等于> b <大于或等于> 0，则对于r（*）（a^<r/> a^pa^<r/>）^<1/> <1/> <1/> <1/> <大于或等于>（a^<r/> b^pa^<r/>）^pa^<r/>）^<1） p+r。 Furuta不平等产生了著名的Lowner-Heinz One（1934），即A <大于或等于> B <大于或等于> 0，可确保a^p <大于或等于> b^p，对于1 <或等于或等于> p <或等于> p <或等于或等于> 0时，当我们将r = 0放在（*IN）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。我们引用了其中一些如下：（Alpha_1）相对操作员熵，（Alpha_2）Ando-Hiai log Marparization，（Alpha_3）Aluthge Transformation，（B_1）Heinz-Kato不平等，（B_2）Kosaki Trace Intoreality，（C_1）PEDERSEN-PEDERSEN-TACERSEN-TACESAKI-TAKESAKIAKI operator Equationatortorator equation。最近，我们获得了一页的一页简化证明，证明了这种广义的Furuta不平等和相关操作员功能中普遍的不平等关系和等效关系。我们必须考虑坎特罗维奇不平等，持有人麦卡锡不平等，里德不平等，詹森不平等和弗鲁塔不平等之间的某些关系。预计Furuta不平等的进一步应用和发展