Review of Bounded Linear Operators of Modules

Abstract

The aim of our work is review of modules and bounded linear operators (B.L.O). Assume that V is a vector space (v.s.) over a field F. Put T is a linear operator. Put R=F[x] is the ring of polynomials in x with coefficients in F. Define ∅:R×V⟶V by ∅(P,v)=P(T)v=P.v ..That ∅ makes V a left R-module denoted V_T. The generalization of this concept have been introduced, put V is a normed space over a field F, put T is a B. L. O. , and assume that R=F[x,y] is the ring of polynomials in x,y with coefficients in F. Define Ψ:R×V→V by Ψ(P,v)=P.v=P(T,T^* )v . 〖 V〗_(T,T^* ) is module. Some properties of thes concepts have been studied.