Establishment and its application of operator inequalities by using computer

计算机算子不等式的建立及其应用

• 批准号: 06640273
• 负责人: FURUTA Takayuki
• 金额: $ 1.34万
• 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: 1994
• 资助国家: 日本
• 起止时间: 1994 至 1996
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-06640273/
• 关键词: Lowner-Heinz inequality; Furuta inequality; relative operator entropy; chaotic order; log majorization; order preserving inequality; log majorization; The Furuta inguality; Lowner-Heinz therkm; relative operatov entropy; positive operaton; trac

In what follows, a capital letter means a bounded linear operator on a complex Hilbert space H.The very famous Lowner-Heinz theorem (1934) asserts that if A<greater than or equal>B<greater than or equal>0 ensures A^<alpha><greater than or equal>B^<alpha> for any alpha <not a member of> [0,1]. But A<greater than or equal>B<greater than or equal>0 does not always hold A^p<greater than or equal>B^p for p>1 in general. As an extension of the Lowner-Heinz theorem, we established the Furuta Inequality (1987) which reads as follows. If A<greater than or equal>B<greater than or equal>0, then for each r<greater than or equal>0 (A^rA^pA^r)^<1/q><greater than or equal>(A^rB^pA^r)^<1/q> hold for p<greater than or equal>0 and q<greater than or equal>1 with (1+2r) q<greater than or equal>p+2r. The Furuta inequality yieldRecently we obtained a lot of applications by using Furuta inequality in the following three branches (a) operator inequalities, (b) norm inequalities and (c) operator equations. We cite some of them as follows : (alpha_1) Applications to relative operator entropy, (alpha_2) Applications to Ando-Hiai log majorization, (alpha_3) Generalized Aluthge transformation, (b_1) Heinz-Kato inequality, (b_2) Kosaki trace inequality, (c_1) Pedersen-Takesaki operator equation.

在接下来的内容中，大写字母是指在复杂的希尔伯特太空中的有限线性操作员。但是a <大于或等于> b <大于或等于> 0，并不总是保持a^p <大于或相等> b^p，对于p> 1而言。作为Lowner-Heinz定理的扩展，我们建立了Furuta不平等（1987），如下所示。如果a <大于或等于> b <大于或等于> 0，则对于每个r <大于或等于> 0（a^ra^pa^r）^<1/q> <1/q> <大于或等于>（a^rb^pa^r）^<1/q>在p <大于或等于> 0和q <大于或等于（1+2r）q <或等于（1+2r）q <> p <2r> p <2 r> p <2 r> p <> p <> p <ofer>> p <imes> p <agips> p <a Furuta不等式屈服于我们通过在以下三个分支（a）操作员不等式的情况下使用furuta不等式获得了许多应用程序，（b）规范不平等和（c）操作员方程。 We cite some of them as follows : (alpha_1) Applications to relative operator entropy, (alpha_2) Applications to Ando-Hiai log majorization, (alpha_3) Generalized Aluthge transformation, (b_1) Heinz-Kato inequality, (b_2) Kosaki trace inequality, (c_1) Pedersen-Takesaki operator equation.

项目成果

T.Furuta: "Complements to the Furuta inequality, III,(with M.Fujii and E.Kamei)" Math.Japonica. 45. 25-32 (1997)

T.Furuta：“Furuta 不等式的补充，III，（与 M.Fujii 和 E.Kamei）”Math.Japonica。

T. Furuta: "Extension of the Furuta inequality and Ando-Hiai log majorijation" Linear Algebra and Its Applications. 219. 139-155 (1995)

T. Furuta：“Furuta 不等式的扩展和 Ando-Hiai 对数主数”线性代数及其应用。

T.Furuta: "Generalizations of Kosaki trace inequalities and related trace inequalities on chaotic order" Linear Alg.and Its Appl.235. 153-161 (1996)

T.Furuta：“Kosaki 迹不等式和混沌序相关迹不等式的概括”线性算法及其应用235。

T.Furuta: "Parallelism related to the inequality "A<greater than or equal>B<greater than or equal>0 ensures (A^<r/2>A^pA^<r/2>)^<(1+r)/><greater than or equal>(A^<r/2>B^pA^<r/2>)^<(1+r)/> for p<greater than or equal>1 and r<greater than or equal>0"" Math.

T.Furuta：“与不等式“A<大于或等于>B<大于或等于>0 相关的并行性确保 (A^<r/2>A^pA^<r/2>)^<(1 r

T.Furuta: "An extension of the Furuta inequality and Ando-Hiai log majorization" Linear Alg.and Its Appl.219. 139-155 (1995)

T.Furuta：“Furuta 不等式和 Ando-Hiai 对数优化的扩展”线性算法及其应用 219。