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

${\displaystyle \int _{\Omega }\int _{\Omega }|k(x,y)|^{2}\,dx\,dy<\infty }$

(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

${\displaystyle (Ku)(x)=\int _{\Omega }k(x,y)u(y)\,dy.}$

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

${\displaystyle \Vert K\Vert _{\mathrm {HS} }=\Vert k\Vert _{L^{2}}.}$

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

${\displaystyle (Kf)(x)=\int _{X}k(x,y)f(y)\,dy}$

is compact. If

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

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.