距離空間内の勾配流とその応用

在这一学年的研究人员继续工作扩大Loewner定理的几个变量。第一个结果是算子凹函数和单调函数的一个新的LMI刻画。这就刻画了函数是辅助Hilbert空间上线性矩阵不等式的唯一极值解。这个结果在RIMS的“Recent developments in operator algebras”研讨会上发表，后来利用这个LMI表示建立了一个精确解公式，这似乎是获得Loewner定理解析延拓部分的关键。这部分目前仍在进行中。作为副产品，无限维版本的矩阵凸集理论和非交换哈恩-巴拿赫定理被建立为关键工具。

In this academic year the researcher continued the work on extending Loewner’s theorem to several variables. The first result is an new LMI characterization of operator concave and monotone functions. This characterizes the functions as unique extremal solutions of linear matrix inequalities over some auxiliary Hilbert space. This result was presented at the ’Recent developments in operator algebras’ workshop in RIMS.Later using this LMI representation an exact solution formula was established which seems to be the key to obtain the analytic continuation part of Loewner’s theorem. This part is still being carried out at the moment. As a by-product an infinite dimensional version of the theory of matrix convex sets and non-commutative Hahn-Banach theorems were established as key tools.