Study on operator theory and operator means

算符平均可以由一定的标量平均以自然的方式构造，而这种算符平均的么正不变范数的比较需要相应的标量平均的比率的正定性。确定了许多这类比率的正定性，从而得到了许多非常精确的范数不等式(如Heinz不等式的大量推广和算子二项平均范数参数的单调性)。此外，许多以这种方式产生的正定函数被证明是无穷可除的。在von Neumann代数的背景下，证明了Jensen型优超结果和迹Jensen不等式的等价条件。

Operator means can be constructed from certain scalar means in a natural fashion, and comparison of unitarily invariant norms of such operator means requires positive definiteness of ratios of corresponding scalar means. Positive definiteness for many such ratios was determined, and consequently many very precise norm inequalities (such as an abundance of generalizations of the Heinz inequality and monotonicity in a parameter of norms of operator binomial means) were obtained. Moreover, many such positive definite functions arising in this way were shown to be infinitely divisible. Also, Jensen-type majorization results and the equality condition for the trace Jensen inequality were proved in the setting of von Neumann algebras.