When does a kernel generate a nuclear operator?

Abstract

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) = ∫₀^a(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.