On the nuclearity of integral operators

Abstract

Let X be a nonempty measurable subset of $\mathbb{R}^m$ and consider the restriction of the usual Lebesgue measure σ of $\mathbb{R}^m$ to X. Under the assumption that the intersection of X with every open ball of $\mathbb{R}^m$ has positive measure, we find necessary and sufficient conditions on a L²(X)-positive definite kernel $K : X \times X \rightarrow \mathbb{C}$ in order that the associated integral operator $\mathcal {K} : L^2(X) \rightarrow L^2(X)$ be nuclear. Taken nuclearity for granted, formulas for the trace of the operator are derived. Some of the results are re-analyzed when K is just an element of $L^2(X \times X)$.