Traceable integral kernels on countably generated measure spaces. (English) Zbl 0724.47014

Brislawn, Chris

Pac. J. Math. 150, No. 2, 229-240 (1991).

Let K be a trace class operator on \(L^ 2(X,{\mathcal M},\mu)\) with integral kernel \(K(x,y)\in L^ 2(X\times X,\mu \times \mu)\). An averaging process is used to define K on the diagonal in \(X\times X\) so that the trace of K is equal to the integral of K(x,x), generalizing results known previously for continuous kernels. This is also shown to hold for positive-definite Hilbert-Schmidt operators, thus giving necessary and sufficient conditions for the traceability of positive integral kernels and generalize previous results obtained by the author using Hardy-Littlewood maximal theory when \(X\subset {\mathbb{R}}^ n\).

Reviewer: Ch.Brislawn

MSC:

47B38	Linear operators on function spaces (general)
47B10	Linear operators belonging to operator ideals (nuclear, \(p\)-summing, in the Schatten-von Neumann classes, etc.)
47G10	Integral operators

Keywords:

singular values; trace class operator; continuous kernels; positive- definite Hilbert-Schmidt operators; traceability of positive integral kernels; Hardy-Littlewood maximal theory