模恒为1的并非常值的向量值解析函数

NON-TRIVIAL VECTOR-VALUED HOLOMORPHIC FUNCTION WITH RANGE IN UNIT SPHERE

称ｆ：Ｕ→Ｅ为向量值解析函数（这里Ｕ为复平面上开集，Ｅ为复Ｂａｎａｃｈ空间），如果对于任意的Ｅ对偶空间中的元素有ｆ为复值解析函数［１］．单复变中最大模原理指出：在内点达到最大模的解析函数，必为常值映射。该相应结论在向量值解析函数中不成立［１］，所以有必要研究模恒为１的并非常值的向量值解析函数。为此，作者引进了Ｂａｎａｃｈ空间单位球面上三种等价关系，利用它们等价类为凸集的性质，以及它们与模恒为１的向量值解析函数的关系，导出一系列结论。其中较有趣的是：ｆ：Ｕ→Ｅ为模恒为１的并非常值的向量值解析函数，且ｆ（Ｕ）Ｅ的有限维子空间，则存在ｅ０，ｅ∈Ｅ－｛θ｝（θ为Ｅ的零元），使得‖ｅ０＋ｚｅ１‖＝１，｜ｚ｜＜１。

Principle of maximum-norm is invalid in the theory of vector-valued holomorphic function[1],so it is necessary to research non-trivial vector-valued holomorphic function with range in unit sphere.In this paper,author introduces three equivalent relations in unit sphere of Banach space,finds out the relation between the above funtion and the equivalent class,and get some interesting conclusions.One of them is that if there exists a function which is non-trivial vector-valued holomorphic function with its range in unit sphere of some finite dimensional subspace,then there exist e0,e1∈E-{θ}(θ is the neutral element) satisfying ‖e0+ze1‖=1 for any z∈C with |z|<1.