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
(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
Then K is a Hilbert–Schmidt operator with Hilbert–Schmidt norm
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
is compact. If
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.
- Renardy, Michael and Rogers, Robert C. (2004). An introduction to partial differential equations. Texts in Applied Mathematics 13 (Second edition ed.). New York: Springer-Verlag. p. 356. ISBN 0-387-00444-0. (Sections 7.1 and 7.5)