Banach空间中Daugavet算子方程的解的存在性

定义了两种子：（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‖的Ｔ的特征值。

The Existence of Solutions to the Daugavet Operator Equation in Banach Spaces

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.