Several Basic Results Of Almost Surely Bounded Linear Operators In Complete RN Spaces

Under a new version of random metric theory,this paper improved and proved the conclusion that if (S₁, X¹) and (S₂, X²) are two RN spaces over the scalar field K with base (Ω, A, μ),then (B(S₁,S₂),X) is complete whenever (S₂, X²) is complete,where B(S₁,S₂) is the linear space of all almost surely (briefly,a.s.) bounded linear operators from S₁ to S₂, and (B(S₁, S₂), X) is the RN space formed by B(S₁,S₂). And by making full use of the completeness of (B(S₁,S₂),X) we proved that when T is an a.s.bounded linear operator in complete random normed spaces and μ({ω∈Ω : X_T(ω)≥ 1}) = 0, the operator (I - T) has an a.s. bounded inverse operator. In addition ,the spectrum of a.s. bounded linear operators in complete random normed modules was introduced.The essential difficulties in studying the spectrum were pointed out.We first introduced the development of random metric theory,especially some important results which Chinese scholars have obtained in recent years, then we simply introduced the notion which would be used in the paper.