完备随机内积模中的Friedrichs定理

完备随机内积模是Hilbert空间的随机推广.最近,经典的Riesz表示定理已经被推广到完备随机内积模上,在此基础上本文将Hilbert空间上经典的Friedrichs定理推广到完备随机内积模上.首先,证明完备随机内积模上任一正Hermite型惟一地对应一个正自共轭算子.值得指出的是:完备随机内积模上Friedrichs定理的证明中所涉及的一系列基本概念与方法都是以随机共轭空间理论为出发点的,与经典情形完全不同.

Friedrichs Theorem in Complete Random Inner Product Modules

The notion of a complete random inner product module is a random generalization of that of a Hilbert space. Recently, the classical Riesz representation theorem on Hilbert spaces has been generalized onto complete random inner product modules, in this paper the classical Friedrichs theorem on Hilbert spaces is generalized onto complete random inner product modules. First,it is proved that a positive Hermite form corresponds uniquely to a positive self-adjoint operator in a complete random inner product module, then this result leads directly to the proof of the main result of this paper. It should be pointed out that the main result of this paper and its proof were completely based on the viewpoint of the theory of random conjugate spaces, and thus the work of this paper had a new starting point which was completely different from that of the corresponding classical case.