Hilbert–Schmidt integral operator

In mathematics, a Hilbert–Schmidt integral operator is a type of integral transform. Specifically, given a domain Ω in Rⁿ, any k : Ω × Ω → C such that

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

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

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

is called a Hilbert–Schmidt integral operator.¹² 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.³

The concept of a Hilbert–Schmidt integral operator may be extended to any locally compact Hausdorff space X equipped with a positive Borel measure. If L²(X) is separable, and k belongs to L²(X × X), then the operator T : L²(X) → L²(X) defined by

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

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.⁴