Research on Inequalities for Operators

本研究旨在研究数学和物理中出现的算子和映射之间的不等式，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。在本研究中，T.Ando和M.D.Choi，以及F.HIAI和Y.Nakamura充分分析了非线性完全正映射的结构。在此基础上，T.Ando和W.Szymanski建立了Lebesgue型分解定理，除了基于序结构的观测外，对于估计扰动算子的特征值和奇异值，优超的概念是相当强大的。在本研究中，T.Ando和R.Bhatia分析了有限维情况。更进一步，T.Ando，R.Horn和C.Johnson建立了Hadamard乘积的控制不等式的完整描述，在无限维的情况下，定义了任意有界线性算子的S数。F.Haii和Y.Nakamura得到了S数在未扰动算子和扰动算子之间的基本优超不等式，Y.Nakamura研究了一维扰动的一个特殊性质，K.Takahashi研究了压缩的拟相似问题，当算子被一个递增的条件期望序列逼近时，收敛总是有问题的.在适当的限制下，F.HIAI和M.Tsukasa在von Neumann代数的框架下解决了这一收敛问题。

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.