When does a kernel generate a nuclear operator?

Let K ∈ L∞ ([0, 1]2) be such that for λ-almost all t ∈ [0, 1] the function K (t, ·) is continuous, ɑ : [0, 1] → [0, 1] a continuous bijective function and U : C[0, 1] → C [0, 1] the operator defined by (Uƒ) (x) = ∫0a(x) ƒ(t)K (t, x) dt: We prove that U is compact and absolutely summing, but U is nuclear if and only if K (t, a-1 (t)) = 0 for λ-almost all t ∈ [0, 1] .Keywords: Banach spaces, continuous functions, compact, nuclear, p-summing