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

[math]\displaystyle{ \int_{\Omega} \int_{\Omega} | k(x, y) |^{2} \,dx \, dy \lt \infty }[/math]

(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

[math]\displaystyle{ (K u) (x) = \int_{\Omega} k(x, y) u(y) \, dy. }[/math]

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

[math]\displaystyle{ \Vert K \Vert_\mathrm{HS} = \Vert k \Vert_{L^2}. }[/math]

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

[math]\displaystyle{ (Kf)(x) = \int_{X} k(x,y)f(y)\,dy }[/math]

is compact. If

[math]\displaystyle{ k(x,y) = \overline{k(y,x)} }[/math]

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.

See also

• Hilbert–Schmidt operator

References

• Renardy, Michael; Rogers, Robert C. (2004). An introduction to partial differential equations. Texts in Applied Mathematics 13 (Second ed.). New York: Springer-Verlag. pp. 262. ISBN 0-387-00444-0.  (Sections 8.1 and 8.5)

• Bump, Daniel (1998). Automorphic Forms and Representations. Cambridge Studies in Advanced Mathematics. 55. Cambridge: Cambridge University Press. pp. 168. ISBN 0-521-65818-7.