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Kernels of trace class operators. (English) Zbl 0695.47017

Summary: Let \(X\subset {\mathbb{R}}^ n\) 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 Duffo 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 \(L^ 2({\mathbb{R}}^ n)\).

MSC:

47B10 Linear operators belonging to operator ideals (nuclear, \(p\)-summing, in the Schatten-von Neumann classes, etc.)
47B38 Linear operators on function spaces (general)
22E27 Representations of nilpotent and solvable Lie groups (special orbital integrals, non-type I representations, etc.)
47A70 (Generalized) eigenfunction expansions of linear operators; rigged Hilbert spaces
47L90 Applications of operator algebras to the sciences
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References:

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