Research of operator inequalities and log-hyponormal operators

本研究的目的是发展算子不等式理论，研究对数次正规算子及相关算子类的性质。由于K.Tanahashi和a.u uchiyama的主要结果如下：如果算子A是超常或A（s,t）类，则谱上一个孤立点所对应的Riesz幂等，如果该点不为0，则该点是自伴随的。如果A是A（s,t）类算子，则Riesz幂等等于它的Aluthge变换的Riesz幂等。当且仅当A和B都是log-次非正常（resps .class A(s,t)）时，A， B的张量积是log-次非正常（resps .class A(s,t)）。Fugled-Putnam型定理适用于p-次正规、p-拟次正规算子。对于内射(p,k)-拟次正规算子，Fugled-Putnam型定理成立。Fugled-Putnam类型定理适用于对数次正规算子或Y类算子。代数上的超常算子满足Wyle定理。Putnam类型不等式适用于A类操作符。M.Takemoto， A.Uch . More iyama和L.Zsido证明了Rn或Cn的任何凸子集都是σ-凸的。这为希尔伯特空间算子的数值范围的参数提供了一个有用的作用。Y.Miura利用Hilbert空间上与标准von Neumann代数相关的完全正映射的概念，证明了算子代数理论中Jordan同构分解定理的非交换L2版的Kadison定理。他在自己的联合著作中引入了Hilbert空间中自对偶锥上所有有界线性算子集合在自对偶锥上的正映射意义上的偏序，并研究了重要锥上该阶的几个性质，如算子的幂以及与所有厄米算子上通常定义的阶的区别。他还共同研究了由x_{n+l}=f（x_{n-1},x_n）定义的递归序列的收敛性问题，并给出了比史提夫克等人的结果更简单、更普遍的肯定答案，并将其应用于生物数学中的物种竞争模型。少

The aim of this research is to develop the theory of operator inequality and study the properties of log-hyponormal and related classes of operators. The main results due to K.Tanahashi and A.Uchiyama is as follows ; If an operator A is paranormal or class A(s,t), then Riesz idempotent corresponding to an isolated point of spectrum is self-adjoint if the point is not 0. If A is a class A(s,t) operator, then the Riesz idempotent is identical to the Riesz idempotent of its Aluthge transform. A tensor product of A, B is log-hyponormal (resp.class A(s,t)) if and only if both A and B are log-hyponormal (resp.class A(s,t)). Fugled-Putnam type theorem holds for p-hyponormal, p-quasihyponormal operators. Fugled-Putnam type theorem holds for injective (p,k)-quasihyponormal operators. Fugled-Putnam type theorem holds for log-hyponormal operators or class Y operators. Algebraically paranormal operators satisfies Wyle's theorem. Putnam type inequality holds for class A operators. M.Takemoto, A.Uch … More iyama and L.Zsido proved that any convex subsets of Rn or Cn is σ-convex. This gives a useful role for the arguments of the numerical ranges of Hilbert space operators. Y.Miura proved the non-commutative L2 version of a Kadison theorem on the decomposition theorem of Jordan isomorphisms in the theory of operator algebras using the notion of completely positive maps on the Hilbert space associated with a standard von Neumann algebra. He introduced in his joint works the partial order on the set of all bound ed linear operators on a Hilbert space with a selfdual cone in the sense of positive maps on selfdual cones, and investigated several properties of this order for the important cones such as powers of operators and the difference with the usual order defined on all hermitian operators. He also as joint worksdealed with the convergency problem of a recursive sequences defined by x_{n+l}=f(x_{n-1},x_n), and gave the affirmative answer more simply and generally than the results of Stevic and his fellows, and applied to a model for competing species in biomathematics. Less