Hilbert–Schmidt integral operator

In mathematics, a Hilbert–Schmidt integral operator is a type of integral transform. Specifically, given a domain (an open and connected set) Ω in n-dimensional Euclidean space Rn, a Hilbert–Schmidt kernel is a function k : Ω × Ω → C with

\int_{\Omega} \int_{\Omega} | k(x, y) |^{2} \,dx \, dy < \infty

(that is, the L2(Ω×Ω; C) norm of k is finite), and the associated Hilbert–Schmidt integral operator is the operator K : L2(Ω; C) → L2(Ω; C) given by

(K u) (x) = \int_{\Omega} k(x, y) u(y) \, dy.

Then K is a Hilbert–Schmidt operator with Hilbert–Schmidt norm

\Vert K \Vert_\mathrm{HS} = \Vert k \Vert_{L^2}.

Hilbert–Schmidt integral operators are both continuous (and hence bounded) and compact (as with all Hilbert–Schmidt operators).

The concept of a Hilbert–Schmidt operator may be extended to any locally compact Hausdorff spaces. Specifically, let X be a locally compact Hausdorff space equipped with a positive Borel measure. Suppose further that L2(X) is a separable Hilbert space. The above condition on the kernel k on Rn can be interpreted as demanding k belong to L2(X × X). Then the operator

(Kf)(x) = \int_{X} k(x,y)f(y)\,dy

is compact. If

k(x,y) = \overline{k(y,x)}

then K is also self-adjoint and so the spectral theorem applies. This is one of the fundamental constructions of such operators, which often reduces problems about infinite-dimensional vector spaces to questions about well-understood finite-dimensional eigenspaces. See Chapter 2 of the book by Bump in the references for examples.

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References

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