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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.古田发现了一个非常有趣的算子不等式，它是Lowner-Heinz不等式的推广。现在这个不等式被称为古田不等式，并得到了许多推广和应用。T.古田证明了包含古田不等式和Ando-Hiai不等式的大古田不等式。所以，M。Fujii证明了许多关于混沌序的不等式。A. Alfrethge利用古田不等式研究p-亚正规算子，并成功地证明了p-亚正规算子的许多有趣的性质。Tanahashi证明了大Furuta不等式的最佳可能性，并证明了Furuta不等式对具有A的Banach ^*-代数成立。内山Tanahashi证明了Putnam不等式和角切割性质对对数亚正规算子成立。另外，塔纳哈什 ...更多信息证明了p-亚正规、p-拟亚正规、对数亚正规A类算子谱的非零孤立点的Riesz幂等元与M. Cho，A.内山Tanahashi证明了施瓦茨型算子不等式，它是Heinz-Kato不等式与A. Uchiyama和M.内山Tanahashi用M. Cho.Takemoto利用von Neumann代数的预对偶空间引入了任意von Neumann代数的元素的数值值域的概念，并证明了以下性质：第一个结果是，所引入的作用在Hilbert空间上的von Neumann代数的任意元素的数值值域的概念等价于Hilbert空间上的任意算子的数值值域的通常概念。第二个结果是利用上述结果证明了每个算子的数值值域在由元素生成的von Neumann代数的^*-同构下是不变的.Miura在具有自对偶锥的复Hilbert空间上引入了所有有界算子集合上的偏序，使得两个算子的差保持锥，并证明了Radon-标准Hilbert空间中算子的Nikodym型定理.此外，Miura从完全正性的角度证明了Hilbert-Schmidt类算子在矩阵序标准型中的优控性质。Miura还证明了矩阵序von Neumann代数之间的不一定保持^* 的同态和底层Hilbert空间之间的完全序同态是相互对应的，通过应用Schmitt-Wittstock工作中的非交换L2-空间的特征。特别地，保^* 同态对应于正交分解同态。少