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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 

 

Kernels of trace class operators


Author: Chris Brislawn
Journal: Proc. Amer. Math. Soc. 104 (1988), 1181-1190
MSC: Primary 47B38; Secondary 47B10
DOI: https://doi.org/10.1090/S0002-9939-1988-0929421-X
MathSciNet review: 929421
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Abstract: Let $ X \subset {{\mathbf{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 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$ 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}({{\mathbf{R}}^n})$.


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DOI: https://doi.org/10.1090/S0002-9939-1988-0929421-X
Article copyright: © Copyright 1988 American Mathematical Society