Research on Inequalities for Operators

本文主要研究数学和物理中算子与映射之间的不等式，直到最近，线性算子空间之间的映射的完全正性仅被考虑为线性映射。本文利用T。安藤&医学博士Choi和F. Hiai & Y.中村。进一步T。Ando & W. Szymanski建立了Lebesgue型分解定理，为了估计扰动算子的特征值和奇异值，除了基于序结构的观测外，优控制的概念是非常强大的。在本研究中，T.安藤河Bhatia分析了有限维情况。进一步T。安多河Horn & C.约翰逊等人建立了Hadamard积的控制不等式的完整描述，在无限维情形下，对任何有界线性算子定义了s-数。F. Hiai & Y.中村得到了s-数在扰动算子和未扰动算子之间的基本控制不等式，Y.中村等人的研究，而K.当一个算子用条件期望的递增序列逼近时，收敛性总是有问题。在适当的限制下，F. Hiai & M.冢司

This research aims at the investigation of inequalities among operators and maps which appear in mathematics and physics.Until recently the complete positivity of a map between the spaces of linear operators has been considered only for a linear map. In the present research the structure of a non-linear completely positive map is fully analyzed by T. Ando & M.D. Choi, and F. Hiai & Y. Nakamura. Further T. Ando & W. Szymanski established a Lebesgue type decomposition theorem.To estimate eigenvalues and singular values of a perturbed operator, beside an observation based on order structure, the notion of majorization is quite powerful. In the present research T. Ando & R. Bhatia analyzed the finite dimensional case. Further T. Ando, R. Horn & C. Johnson have settled a complete description of majorization inequalities for Hadamard products.In the infinite dimensional case, s-numbers are defined for any bounded linear operator. F. Hiai & Y. Nakamura obtained fundamental majorization inequalities for s-numbers between an unperturbed and a perturbed operator.A special character of one-dimensional perturbation was studied by Y. Nakamura while the problem of quasi-similarity of contractions was pursued by K. Takahashi.When an operator is approximated by an increasing sequence of conditional expectations, the convergence always comes into problem. Under suitable restrictions, this convergence was settled in the frame of von Neumann algebras by F. Hiai & M. Tsukasa.