Vol. 99, No. 1, 1982

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Characterization and order properties of pseudo-integral operators

Ahmed Ramzy Sourour

Vol. 99 (1982), No. 1, 145–158
Abstract

Let (X,𝒜,m1) and (Y,,m2) be separable σ-finite measure spaces. A linear transformation T from an order-ideal L of measurable functions on Y into the space of measurable functions on X is called a pseudo-integral operator if it is induced by a measure μ on X × Y via the equation

∫                    ∫ ∫
(Tf)(x)g(x)m (dx) =     f(y)g(x)μ(dx,dy)
1

for sufficiently many functions g. Our main theorem states that T is a pseudo-integral operator if and only if Tfn 0 a.e. whenever 0 fn f L and fn 0 a.e. We also study the order structure of the class of pseudo-integral operators showing that they form a band (order-closed ideal) in the space of order-bounded operators.

Mathematical Subject Classification 2000
Primary: 47B38
Milestones
Received: 3 March 1980
Revised: 30 July 1980
Published: 1 March 1982
Authors
Ahmed Ramzy Sourour