完备随机赋范空间中几乎处处有界线性算子的几个基本结果

Several Basic Results of Almost Surely Bounded Linear Operators in Complete Random Normed Spaces

在随机度量理论的新版本下,改进并重新证明了如下结论:设(S1,X1)和(S2,X2)均为数域K上以(Ω,A,μ)为基的随机赋范空间,当S2是完备时,(B(S1,S2),X)亦为完备的,其中(B(S1,S2),X)为所有定义在S1上取值于S2中的几乎处处(简写为a.s.)有界线性算子所成的随机赋范空间.并在此基础上证明了当T为完备随机赋范空间S上a.s.有界线性算子时,如果μ({ω∈Ω:XT(ω)≥1})=0,则算子I-T有a.s.有界逆算子.此外还引入了在完备随机赋范模中几乎处处有界线性算子的谱的概念,并指出关于这种谱研究中的本质困难.

Under the new version of random metric theory, this paper improved and proved the conclusion that if （S2, X2 ） and （S2 ,X2） are two Random Normaled 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 , Sz ） 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 Random Normed spaces 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,XT（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 weté pointed out.