Hilbert–Schmidt integral operator
In mathematics, a Hilbert–Schmidt integral operator is a type of integral transform. Specifically, given a domain Ω in n-dimensional Euclidean space Rn, a Hilbert–Schmidt kernel is a function k : Ω × Ω → C with
, and the associated Hilbert–Schmidt integral operator is the operator K : L2 → L2 given by
Then K is a Hilbert–Schmidt operator with Hilbert–Schmidt norm
Hilbert–Schmidt integral operators are both continuous and compact.
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 is a separable Hilbert space. The above condition on the kernel k on Rn can be interpreted as demanding k belong to L2. 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.
