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, a : [0, 1] → [0, 1] a continuous bijective function and U : C[0, 1] → C [0, 1] the operator defined by (U f ) (x) = ∫ ₀^a(x) f (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].