Let E and F be two Banach
function spaces on two σ-finite measure spaces (Y,Σ,ν) 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) =
∫YK(x,y)f(y)dν(y) for μ-almost all x, and
∫Y|K(⋅,y)f(y)|dν(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.
