Banach空间中Daugavet算子方程的解的存在性

摘要

Two kinds of operators are defined:the operator of type(Ⅰ)and that of type(Ⅱ).The following theorem is proven:If a linear continuous operatorT:X→X on a Banach space X is of type(Ⅰ)or type(Ⅱ),then T satisfies the Daugavet equation ‖I+T‖=1+‖T‖if and only if the norm of the operator is an eigenvalue of T.Applications of this result are presented.For example,it is asserred that a compact operator T:X→X on a weakly locally uniformly convex Banach space X satisfies the Daugavet equation if and only if its norm ‖T‖is an eigenvalue of T.%定义了两种算子:(Ⅰ)型算子与(Ⅱ)型算子.证明了下列定理:若Banach空间X上线性连续算子T:X→X是(Ⅰ)型算子或(Ⅱ)型算子,则T满足Daugavet方程‖I+T‖=1+‖T‖的充要条件是算子T的范数‖T‖是T的特征值.另一方面,给出了该结果的应用.例如,由此断言,弱局部一致凸Banach空间X上紧算子T:X→X满足Daugavet方程的充要条件是范数‖T‖是T的特征值.