【作者】 汤约翰；

【导师】 胡俊云；

【作者基本信息】 湘潭大学 ， 基础数学， 2004， 硕士

【摘要】 本文在第一章中给出了一些预备知识，内容主要涉及到郭铁信所提出的随机泛函分析的一些基本概念：RN空间；RN空间上随机算子与随机泛函的a．s．有界；RN模及其完备化(我们称之为RB模)；RIP空间，RIP模及其完备化(我们分别称之为RH空间，RH模)：另外还给出了已经证明的RH模上a．s．有界随机线性泛函的Riesz表示定理。 在第二章，定义了随机Banach代数以及其中元素的谱，并研究了其基本性质，将经典Banach代数理论中的许多结果推广到了随机Banach代数中。为了给出一个合适的谱的定义，我们进行了许多尝试。如果不注意到随机变量本身的特性，而是直接用经典的谱定义，那么许多结果变的很坏，甚至没有任何的规律性。举例来说，我们就看L(Ω，C)，这个特殊的随机Banach代数，考察随机变量(?)其中A∈σ且P(A)=1/2。这是一个自伴元，但如果按照经典的谱定义的话，那么它的谱将不会是实的，例如复值随机变量山任A山必AI一2︵11︸r!、l、 一一 、，了 O) 了‘、 斤其中A oa且P(A)二专，它就在其中。从这个例子中我们也初步地看到正测集扮演着重要的角色，这导致我们最终采用了现在的定义。 在第三章，定义了随机C’一代数以及其中的正常元，自伴元，投影，酉元，其上的a.5.有界的随机正线性泛函，算子的a.5.下有界等概念，并证明了代数中酉元和自伴元的谱定理;作为一类特殊的随机C’一代数的B(习，我们利用定理3.2把算子的谱的问题转化为算子的a.5.下有界的问题，并证明了自伴算子和正算子的谱定理。另外在随机C‘一代数上a.5.有界的随机正线性泛函存在的情况下，我们给出了相应的GNS构造。 文〔14〕得到随机内积模上正算子及正交投影算子的一些性质，为了继续研究正算子的性质，在第四章中给出了RH模上与正算子相关的一些不等式，以及由此得出的相关结论，这些结果有利于进一步研究RH模上正算子的性质。更多还原

【Abstract】 In the first chapter of this paper some preliminaries are given. The content is mainly about some basic notions of random functional analysis Guo Tiexin introduced: RN space; a.s. boundedness of random operator and random functional on RN space; RN module and its completion (we call it RB module); RN module and its completion (we call it RB module); RIP space, RIP module and its completion (we separately call them RH space, RH module); moreover, we give the proved Riesz representation theorem about a.s. bounded random linear functional on RH module.In the second chapter, we define random Banach algebra and spectrum of its element, and study its basic properties, and generalize many results in the classical Banach algebra theory to the case in random Banach algebra. In order to give a suitable definition of spectrum, we did much try. If we turn a deaf ear to the property of random variable and use the classical definition of spectrum, thenmany results become badly and even have no regularity. For instance, let us consider L(Q,C), a particular random Banach algebra, inspect a random variablewhere A e a and P(A) = \. This is a self-adjoint element, but if we use the classical definition of spectrum, then its spectrum will not be real. For example, a complex valued random variablewhere A e a and P(A) = j-, it is in it. We initially see the important role the positive measure set plays from this example, which leads us to use the present definition.In the third chapter, we define some notions of random C*-algebra and its normal element, self-adjoint element, projection, unitary element, a.s. bounded random positive linear functional, a.s. bounded below of an operator etc, and prove the spectrum theorem of unitary element and self-adjoint element; as a particular randomC*-algebra B(S), we use theorem 3.2 to transform the problem of spectrum into the problem of a.s. bounded below, and prove the spectrum theorem of self-adjoint operator and positive operator. Moreover, we give the correspondent GNS construction under the case that an a.s. bounded random positive linear functional on a random C*-algebra exists.Some properties of positive operators and orthogonal projection operators on random inner product module have been obtained in the paper [14]. In order to keep on studying the properties of positive operators, we give some inequalities about positive operators on RH module in the fourth chapter. These results will be advantageous to further research in the properties of positive operators on RH module.更多还原