In this paper we are concerned with bounded generalized random linear operators. It is shown that each bounded generalized random linear operator can be seen as a set-valued random variable. The properties of some special bounded generalized random linear operators and the random resolvent set of generalized random linear operators are investigated. | VNU Journal of Science: Mathematics – Physics, Vol. 33, No. 3 (2017) 95-104 Bounded Generalized Random Linear Operators Nguyen Thinh* Department of Mathematics, VNU University of Science, 334 Nguyen Trai, Hanoi, Vietnam Received 03 May 2017 Revised 30 June 2017; Accepted 15 September 2017 Abstract: In this paper we are concerned with bounded generalized random linear operators. It is shown that each bounded generalized random linear operator can be seen as a set-valued random variable. The properties of some special bounded generalized random linear operators and the random resolvent set of generalized random linear operators are investigated. Ke ywo rd s: Random linear operator, random bounded linear operator, generalized random linear mapping, bounded generalized random linear operator, set-value random variable, random resolvent set, random regular value. AMS S u b jec t Cla s si fi ca t io n 2 0 0 0 : Primary 60H25; Secondary 60H05, 60B11, 45R05. 1. Introduction Let X,Y be separable Banach spaces and ( ,Ƒ,P ) be a probability space. By a random mapping (or a random operator) from X to Y we mean a rule that assigns to each element x X a Y-valued random variable (.). Mathematically, a random mapping defined from X to Y is simply a mapping A : X L0 , Y where L0 , Y stands for the space of all Y-valued random variables (.’s). If S = [a,b ] is a interval of the real line then F F t t a ,b is said to be a Y-valued stochastic process. The interest in random mappings has been arouse not only for its own right as a random generalization of deterministic mappings as well as a natural generalization of stochastic processes but also for their widespread applications in other areas. Research in theory of random mappings has been carried out in many directions including random linear mappings which provide a framework of stochastic integral, infinite random matrix (see . [2, 5, 11], [14-19]), random fixed points of .
