Kernels of trace class operators

Abstract

Let X ⊂ R n X \subset {{\mathbf {R}}^n} and let K K be a trace class operator on L 2 ( X ) {L^2}(X) with corresponding kernel K ( x , y ) ∈ L 2 ( X × X ) K(x,y) \in {L^2}(X \times X) . An integral formula for tr K 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 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 ( R n ) {L^2}({{\mathbf {R}}^n}) .