Hilbert–Schmidt integral operator

In mathematics, a Hilbert–Schmidt integral operator is a type of integral transform. Specifically, given a domain in, any such that
is called a Hilbert–Schmidt kernel. The associated integral operator given by
is called a Hilbert–Schmidt integral operator.
is a Hilbert–Schmidt operator with Hilbert–Schmidt norm
Hilbert–Schmidt integral operators are both continuous and compact.
The concept of a Hilbert–Schmidt integral operator may be extended to any locally compact Hausdorff space equipped with a positive Borel measure. If is separable, and belongs to, then the operator defined by
is compact. If
then 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.