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

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
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