Traceable integral kernels on countably generated measure spaces

Abstract

Let K be a trace class operator on L 2 (X, Jί, μ) with integral kernel K(x, y) € L 2 (X x X, μ x μ). An averaging process is used to define K on the diagonal in X 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 HardyLittlewood maximal theory when IcR".

Lattice Trace Operators

“…Following the account of Brislawn [5], we extend the mapping → ∫ Σ ⟨ , ⟩ from the space C 1 ( 2 ( )) of trace class linear operators to a larger class of regular operators by representing by a "regularised" kernel, so that the collection of regular operators for which ∫ Σ ⟨| |, ⟩ < ∞ is a vector sublattice of the Riesz space of regular operators-a property not necessarily enjoyed by the trace class operators. …”

“…The difficulty is addressed by Brislawn [4,5], [1, Appendix D] who shows that, for a trace class operator : 2 ( ) → 2 ( ) with integral kernel , the equality…”

“…The measure is supposed in [5] to be a -finite Borel measure on a second countable topological space Σ and the regularised kernel̃is defined from by averaging with respect to the product measure ⊗ . Extending the result (c) of M. Duflo given above, Brislawn [5,Theorem 4.3] shows that a hermitian positive Hilbert-Schmidt operator is a trace class operator if and only if ∫ Σ̃( , ) ( ) < ∞.…”

“…The regularised kernel̃: Σ × Σ → C of an absolute integral operator is defined by adapting the method of Brislawn [5] to positive operators with an integral kernel. The generalised trace ∫ Σ̃( , ) ( ) may be viewed alternatively as a bilinear integral ∫ Σ ⟨ , ⟩ with respect to the measure : → , ∈ B. Lattice trace operators are employed in the proof of the Cwikel-Lieb-Rosenblum inequality for dominated semigroups [6].…”

Decaimento dos autovalores de operadores integrais positivos gerados por núcleos Laplace-Beltrami diferenciáveis

“…Algumas destas generalizações e aplicações do Teorema de Mercer podem ser encontradas em [4,5,10,12,26,29,42,59,60,68,77,81,82]. Apresentamos aqui uma versão do teorema para o caso em que Xé um espaço topológico primeiro-enumerável e localmente compacto, algo que não encontramos na literatura e que abrange, com folga, nosso contexto esférico.…”