Kernels of trace class operators

Abstract:

Let $X \subset {{\mathbf {R}}^n}$ and let $K$ be a trace class operator on ${L^2}(X)$ with corresponding kernel $K(x,y) \in {L^2}(X \times X)$. An integral formula for tr $K$, proven by Duflo for continuous kernels, is generalized for arbitrary trace class kernels. This formula is shown to be equivalent to one involving the factorization of $K$ into a product of Hilbert-Schmidt operators. The formula and its derivation yield two new necessary conditions for traceability of a Hilbert-Schmidt kernel, and these conditions are also shown to be sufficient for positive operators. The proofs make use of the boundedness of the Hardy-Littlewood maximal function on ${L^2}({{\mathbf {R}}^n})$.