Vol. 26, No. 2, 1968

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A characterization of integral operators on the space of Borel measurable functions bounded with respect to a weight function

Louise Arakelian Raphael

Vol. 26 (1968), No. 2, 361–366
Abstract

Let I be a Borel set of the real line R, C the space of complex numbers, V a σ-algebra of Borel subsets of I, μ a fixed measure on V such that for any bounded set Q V , μ(Q) < , g(λ,p) a nonvanishing complex valued function μ-measurable in λ I such that |g(λ,p)|↑ in p where p belongs to a fixed open interval (a,b), and S the set of μ-measurable functions u from I into C such that |u(λ)g(λ,p)|m for some p depending on u,p (a,b), m 0 and m depending on u, and for all λ I. The purpose of this paper is to prove the following:

Theorem 1. Let c(λ,δ) be a μ × μ-measurable function on I × I. For every function u S the function

      ∫
y(λ) =  c(λ,δ)u(δ)dμ(δ),(λ,δ) ∈ I × I
I

is well defined and y S if and only if for every p (a,b) there exists a q (a,b) such that

∫
|g(λ,q)c(λ,δ)(g(δ,p))−1|dμ(δ) ≦ m
I

for all (λ,δ) I × I and some m 0.

Mathematical Subject Classification
Primary: 47.70
Milestones
Received: 11 August 1967
Published: 1 August 1968
Authors
Louise Arakelian Raphael