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, then the square-integrable function k : Ω × Ω → C belonging to L²(Ω×Ω) such that

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

is called a Hilbert–Schmidt kernel and the associated integral operator T : L²(Ω) → L²(Ω) given by

${\displaystyle (Tf)(x)=\int _{\Omega }k(x,y)f(y)\,dy,\quad f\in L^{2}(\Omega ),}$

is called a Hilbert–Schmidt integral operator. Then T is a Hilbert–Schmidt operator with Hilbert–Schmidt norm

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

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 L²(X) be a separable Hilbert space and X a locally compact Hausdorff space equipped with a positive Borel measure. The initial condition on the kernel k on Ω ⊆ Rⁿ can be reinterpreted as demanding k belong to L²(X × X). Then the operator

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

is compact. If

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

then T 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.