准Hilbert C~*-模上的范数等式(英文)

Norm equalities in pre-Hilbert C~*-modules

目的为了研究‖x+y‖=‖x‖+‖y‖和毕达哥拉斯等式在准HilbertC*-模中成立的充要条件。方法采用了算子论方法进行研究。结果证明了‖x+y‖=‖x‖+‖y‖成立当且仅当存在A上的态使得(〈‖y‖x-‖x‖y,‖y‖x-‖x‖y〉)=0且(〈x,x〉)=‖x‖2或(〈y,y〉)=‖y‖2成立。也给出了准HilbertC*-模中毕达哥拉斯等式成立的充要条件。结论本文的结果对研究准HilbertC*-模中的范数等式非常有用。

Aim To research the necessary and sufficient conditions for ｜｜x＋y｜｜=｜｜x｜｜＋｜｜y｜｜andPythagoras equality in a pre-Hilbert C^＊ -module. Methods Operator theoretic method is used. Results. ｜｜ x＋y｜｜=｜｜x｜｜＋｜｜y｜｜proves tenable if and only if there exists a state φ of the C＊ -algebra A such that φ（〈｜｜y｜｜x-｜｜x｜｜y,｜｜y｜｜x-｜｜x｜｜y〉）=0and either φ（（x,x））=｜｜x｜｜2 or φ（〈y,y〉）=｜｜y｜｜2 The necessary and sufficient conditions for Pythagoras equality in a pre-Hilbert C^＊-module. Conclusion The result is very useful for researching norm equalities in pre-Hilbert C＊ -modules.