Integral representation of linear operators

Abstract

In this paper we obtain necessary and sufficient conditions in order that a linear operator, acting in spaces of measurable functions, should admit an integral representation. We give here the fundamental results. Let (T_i,μ _i) (i=1,2) be spaces of finite measure, and let (T,μ) be the product of these spaces. Let E be an ideal in the space S(T₁,μ ₁) of measurable functions (i.e., from |e₁|⩽|e₂|, e₁∈ S (T₁,μ ₁), e₂∈E it follows that e₁∈E). THEOREM 2. Let U be a linear operator from E into S(T₂,μ ₂). The following statements are equivalent: 1) there exists aμ-measurable kernel K(t,S) such that (Ue)(S)=∫K(t,S) e(t)dμ(t) (e∈E); 2) if 0⩽e_n⩽∈E (n=1,2,...) and e_n→0 in measure, then (Ue_n)(S) →0μ ₂ a.e. THEOREM 3. Assume that the function Ф(t,S) is such that for any e∈E and for s a.e., theμ ₂-measurable function Y(S)=∫Ф(t,S)e(t)dμ ₁(t) is defined. Then there exists aμ-measurable function K(t,S) such that for any e∈E we have ∫Ф(t,S)e(t)dμ ₁(t)=∫K(t,S)e(t)dμ ₁(t)μ ₁a.e.