Research of operator inequalities and log-hyponormal operators

点击翻译按钮获取中文摘要

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