Compactness of integral operators

Let and be locally compact Hausdorff spaces. For every function on and on , we can define a function on by If and , then is a continous function on with a support contained in . Take distinct points and in . We may assume that , so there is a such that and . Take another function such that , then is a function in such that and , which follows that the subset separates points of and vanishes nowhere. By locally compact version of Stone–Weierstrass theorem, we have that is a dense subset of in the topology of uniform convergence.

Suppose , and be precompact open sets in and that contain the projections and of the support of onto and respectively. Given any , the above argument shows that there are finite pairs of and such that Moreover, we can require that and are supported in and respectively, since we can replace and by and where and are functions in and that are equal to on and and vanish outside and respectively. Note that .

Suppose is a Radon measure on and ( ). For every , as is dense in , there is a such that . Assume and are functions like above, then Take , then we have that . This shows that is dense in ( ).

Proof. When where and , then It shows that is a rank-one operator. When is a linear combination of functions , then is a finite-rank operator. When is given by the above theorem, we can approximate by functions in the linear span of in the topology of . The condition implies that the integral operator induced by is bounded can be approximated by that induced by functions in the linear span of and hence is compact. ◻

1. A condition for being dense in : is a Radon measure. (see Proposition 7.9, p217, Folland’s Real Analysis)

2. todo... Check conditions for product measure.

1. Stone–Weierstrass theorem (locally compact spaces).

Suppose is a locally compact Hausdorff space and is a subalgebra of . Then is dense in (given the topology of uniform convergence) if and only if it separates points and vanishes nowhere (i.e., for all there is a such that ).

2. There is a similar result in Exercise 6, p177, Conway’s functional analysis.

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