Traceable integral Kernels on countably generated measure spaces

Abstract

Let K be a trace class operator on L2(X, M, µ) with integral kernel K(x, y) ∈ L2(X × X, µ × µ). An averaging process is used to define K on the diagonal in X × X so that the trace of K is equal to the integral of K(x, x), generalizing results known previously for continuous kernels. This formula is also shown to hold for positive-definite Hilbert-Schmidt operators, thus giving necessary and sufficient conditions for the traceability of positive integral kernels. These results make use of Doob’s maximal theorem for martingales and generalize previous results obtained by the author using Hardy-Littlewood maximal theory when X ⊂ Rn. © 1991 by Pacific Journal of Mathematics.