Research of log-hyponormal operator and the Furuta inequality

对数次正规算子和Furuta不等式的研究

基本信息

批准号：
12640187
负责人：
TANAHASHI Kotaro
金额：
$ 1.22万
依托单位：
Tohoku Pharmaceutical University
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (C)
财政年份：
2000
资助国家：
日本
起止时间：
2000 至 2002
项目状态：
已结题

项目摘要

T. Furuta discovered a very interesting operator inequality, which is an extension of Lowner-Heinz's inequality. Now the inequality is called the Furuta inequality and many generalizations and application has been developed. T. Furuta proved the grand Furuta inequality which contains the Furuta inequality and Ando-Hiai's inequality. Also, M. Fujii proved many inequalities with respect to chaotic order. A. Aluthge used the Furuta inequality to study p-hyponormal operators and succeeded to prove many interesting properties of p-hyponormal operators. The aim of this research is to develop the theory of operator inequality and study the properties of p-hyponormal, log-hyponormal and related classes of operators.In this research, Tanahashi proved the best possibility of grand Furuta inequality and proved that the Furuta inequality holds for Banach ^*-algebra with A. Uchiyama. Tanahashi proved Putnam's inequality and angular cutting property holds for log-hyponormal operators. Also, Tanahash … More i proved that the Riesz idempotent for non-zero isolated point of spectrum of p-hyponormal, p-quasihyponormal, log-hyponormal, class A operators is self-adjoint with M. Cho, A. Uchiyama. Also, Tanahashi proved Schwarz type operator inequalities which are extensions of Heinz-Kato inequality with A. Uchiyama and M. Uchiyama. Also, Tanahashi proved many spectral relations of Aluthge transform with M. Cho.Takemoto introduced a notion of numerical ranges for the elements of any von Neumann algebra by using the predual space of von Neumann algebra and showed the following properties : The first result is that the introduced notion of numerical range for any element of von Neumann algebra acting on a Hilbert space equivalents to the usual notion of numerical ranges for any operator on a Hilbert space. The second result is that by using the aboved mentioned result the numerical range of each operator is invariant under the ^*-isomorphism of the von Neumann algebra generated by the element.Miura introduced the partial order on the set of all bounded operators on a complex Hilbert space with a selfdual cone in the sense that the difference of two operators preserves the cone and showed the Radon-Nikodym type theorem for operators a standard Hilbert space. Moreover, Miura proved the majorization property of operators of Hilbert-Schmidt class in a matrix ordered standard form from the point of view of complete positivity. Miura also proved that a not necessarily ^*-preserving homomorphism between matrix ordered von Neumann algebras and a complete order homomorphism between the underlying Hilbert spaces correspond to each other by applying the characterization of non-commutative L2-spaces in the work of Schmitt-Wittstock. In particular, a ^*-preserving homomorphism corresponds to a orthogonal decomposition homomorphism. Less

T. furuta发现了一个非常有趣的操作员不平等，这是下海因兹的不平等现象的延伸。现在，不平等被称为Furuta不平等，并且已经开发出许多概括和应用。 T. furuta证明了大呋喃不平等，其中包含Furuta不平等和Ando-Hiai的不平等。同样，富士菌在混乱的秩序方面证明了许多不平等现象。 A. Aluthge使用Furuta不等式研究了P-螺旋正常运算符，并成功证明了P-夏季性算子的许多有趣特性。这项研究的目的是发展操作者的不平等理论，并研究了操作员的p- - 螺旋正常，虫质正常和相关类别的特性。在这项研究中，塔纳哈西证明了大呋喃不平等的最佳可能，并证明了furuta不平等的不平等是Banach ^*-Algebra与A. A. Uchiyama所具有的。塔纳希（Tanahashi）证明了普特南（Putnam）的不平等和角度切割属性，可用于对数 - 夏季正常运营商。另外，塔纳哈什（Tanahash）…更多，我证明了riesz dimpotent用于非零孤立点的p- - 甲基频谱，p-季度，p- quasihyponormal，log-hyponormor，A类操作员与M.此外，塔纳哈西（Tanahashi）提供了施瓦茨（Schwarz）类型的操作员不平等，这些不平等是亨氏 - 卡托（Heinz-kato）与A. Uchiyama和M. Uchiyama的不平等。 Also, Tanahashi provided many spectral relations of Aluthge transform with M. Cho.Takemoto introduced a notification of numerical ranges for the elements of any von Neumann algebra by using the predual space of von Neumann algebra and showed the following properties: The first result is that the introduced notification of numerical ranges for any element of von Neumann algebra acting on a Hilbert space equivalents to希尔伯特空间上任何操作员的数值范围通常。第二个结果是，通过使用上述结果，每个运算符的数值范围在元素产生的von Neumann代数的 ^* - 同构下是不变的。希尔伯特空间。此外，从完全积极的角度来看，Miura以矩阵有序的标准形式被证明是希尔伯特 - 史密特类经营者的多数化财产。 Miura还证明，不必要的 ^* - 在矩阵下订购的von Neumann代数与基础希尔伯特空间之间的完全订单同构之间的同态同构，通过在Schmitt-Wittstock的工作中应用非交通性L2空间的表征相互对应。特别是，保存同态的a ^*对应于正交分解同态。较少的