关于KyFan的一个猜想

ON A CONJECTURE FROM KY FAN

设ｆ（ｚ）为单位圆上的规范化的单叶凸像函数，Ｌ（Ｈ）为复Ｈｉｌｂｅｒｔ空间Ｈ上的Ｂａｎａｃｈ算子代数。本文证明了对于Ｌ（Ｈ）中双重可换的真收缩算子Ｔ＿ｉ（ｉ＝１，２），ｆ（Ｔ＿１）与ｆ（Ｔ＿２）的凸线性组合一定具有形式ｆ（Ｔ），此处Ｔ为Ｌ（Ｈ）中的真收缩算子。同时我们通过一简单例子说明＂Ｔ＿ｉ双重可换＂的条件是不能去掉的，这便回答了由ＫｙＦａｎ提出的一个问题。

let f(z)be a nomalized convex univalent function in the unit disc and L(H)denote the Banach operator algebra for the complex Hilbert space H.In this note we show tha for T_i(i=1,2)being proper contractions In L(H)and commuting two-fordly the convex linear combination of f(T_1)aud f(T_2)is the form of f(T)where T is a proper contraction in L(H).A simple example shows that the condition for T_i two-ford commting cannot be omitted,which gives an answer of the question from Ky Fan.