Research of log-hyponormal operator and the Furuta inequality

T.Furuta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Furuta不等式，并得到了许多推广和应用。T.Furuta证明了包含Furuta不等式和Ando-HIAi不等式的广义Furuta不等式。此外，M.Fujii还证明了许多关于混沌秩序的不等式。A.Aluthge利用Furuta不等式研究了p-次正规算子，并成功地证明了p-次正规算子的许多有趣的性质。本研究的目的是发展算子不等式理论，研究p-次正规、对数次正规及相关的算子类的性质。在这项研究中，Tanahashi证明了Grand Furuta不等式的最佳可能性，并证明了Furuta不等式对于具有A.Uchiyama的Banach^*-代数是成立的。Tanahashi证明了对数次正规算子的Putnam不等式和角割性质成立。还有，Tanahash…进一步证明了p-次正规、p-拟次正规、对数次正规A类算子的谱的非零孤立点的Riesz幂等元是与M.Cho、A.Uchiyama自伴的。此外，Tanahashi与A.Uchiyama和M.Uchiyama一起证明了Schwarz型算子不等式，它是Heinz-Kato不等式的推广。此外，Tanahashi与M.Cho.一起证明了Aluthge变换的许多谱关系。Takemoto利用von Neumann代数的预对偶空间引入了任何von Neumann代数元素的数值值域的概念，并展示了以下性质：第一个结果是，引入的作用于Hilbert空间上的von Neumann代数的任何元素的数值值域的概念等价于通常的Hilbert空间上任何算子的数值值域的概念。第二个结果是，利用上述结果，每个算子的数值值域在由该元生成的von Neumann代数的^*-同构下是不变的。Miura在具有自对偶锥的复Hilbert空间上的所有有界算子的集合上引入了偏序，即两个算子的差保持锥，并证明了关于算子的Radon-Nikodym型定理是标准的Hilbert空间。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形中的优超性质。Miura还利用Schmitt-Wittstock关于非交换L2-空间的刻画，证明了矩阵序von Neumann代数之间不一定^*保持的同态与基础Hilbert空间之间的完全序同态是对应的。特别地，保^*同态对应于正交分解同态。较少

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