Research on Inequalities for Operators

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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.