Integral representation of linear operators

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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 (Ti, μi) (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, μ1) of measurable functions (i.e., from |e1|≤|e2|, e1∈ S (T1, μ1), e2∈E it follows that e1∈E). THEOREM 2. Let U be a linear operator from E into S(T2, μ2). The following statements are equivalent: 1) there exists a μ-measurable kernel K(t,S) such that (Ue)(S)=∫K(t,S) e(t)d μ(t) (e∈E); 2) if 0≤en≤∈E (n=1,2,...) and en→0 in measure, then (Uen)(S) →0 μ2 a.e. THEOREM 3. Assume that the function F{cyrillic}(t,S) is such that for any e∈E and for s a.e., the μ2-measurable function Y(S)=∫F{cyrillic}(t,S)e(t)d μ1(t) is defined. Then there exists a μ-measurable function K(t,S) such that for any e∈E we have ∫F{cyrillic}(t,S)e(t)d μ1(t)=∫K(t,S)e(t)d μ1(t)μ1a.e.

Original languageEnglish
Pages (from-to)129-137
Number of pages9
JournalJournal of Soviet Mathematics
Volume9
Issue number2
DOIs
Publication statusPublished - 1 Feb 1978

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  • Mathematics(all)

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