Vol. 100, No. 1, 1982

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Compactness properties of abstract kernel operators

Charalambos D. Aliprantis, Owen Sidney Burkinshaw and M. Duhoux

Vol. 100 (1982), No. 1, 1–22

Let E and F be two Banach function spaces on two σ-finite measure spaces (Y,Σ) and (X,S,μ) respectively. Then an operator T : E F is called a kernel operator, if there exists a μ×ν-measurable real valued function K(x,y) on X ×Y such that for each f E we have

Tf(x) = Y K(x,y)f(y)(y) for μ-almost all x, and
Y |K(,y)f(y)|(y) F.
It is well known that every kernel operator belongs to the band (E F)dd generated by the finite rank operators, and under certain conditions (E F)dd consists precisely of all kernel operators.

In this paper we consider E and F to be two locally convex-solid Riesz spaces. Motivated by the above remarks, we call every operator in (E F)dd that can be written as a difference of two positive weakly continuous operators, an abstract kernel operator. We characterize the abstract kernel operators that map bounded sets onto precompact sets. In the process we generalize known characterizations of compact kernel operators and obtain some interesting new ones.

Mathematical Subject Classification 2000
Primary: 47B55
Secondary: 46A40
Received: 13 October 1980
Revised: 19 May 1981
Published: 1 May 1982
Charalambos D. Aliprantis
Owen Sidney Burkinshaw
M. Duhoux