A bounded linear operatorTon a Hilbert spaceℋis trace class if its singular values are summable. The trace class operators onℋform an operator ideal and in the case thatℋis finite-dimensional, the trace tr(T)ofTis given by∑jajjfor any matrix representation{aij}ofT. In applications of trace class operators to scattering theory and representation theory, the subject is complicated by the fact that ifkis an integral kernel of the operatorTon the Hilbert spaceL2(μ)withμaσ-finite measure, thenk(x,x)may not be defined, because the diagonal{(x,x)}may be a set of(μ⊗μ)-measure zero. The present note describes a class of linear operators acting on a Banach function spaceXwhich forms alatticeideal of operators onX, rather than anoperatorideal, but coincides with the collection of hermitian positive trace class operators in the case ofX=L2(μ).
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