Research of log-hyponormal operator and the Furuta inequality

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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