On compactness of integral operators with essentially bounded kernels between Banach function spaces

Let (, , μ) and (, , ν) be finite measure spaces. We study the compactness of
the integral operator
(K f )(t) =


K(t,τ) f (τ ) dμ(τ ), t ∈ ,
with a kernel K ∈ L∞( × ,ν ⊗ μ), acting from a Banach function space
X = X(, μ) to a Banach function space Y = Y (, ν). Let X = X
(, μ) be
the associate space of X, let (X
)b and Yb be the closures of simple functions in X
and Y , respectively, and let (X
)a and Ya be the subspaces of functions of absolutely
continuous norms of X and Y , respectively. We show that (1) if (X
)a = (X
)b and
Ya = Yb, then K : X → Y is compact; (2) if ν is nonatomic and (X
)a = (X
)b or μ
is nonatomic and Ya = Yb, then there exists a kernel K ∈ L∞( × ,ν ⊗ μ) such
that K : X → Y is noncompact.