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

$\underset{\mathrm{\Omega }}{\int }\underset{\mathrm{\Omega }}{\int }|k(x,y){|}^{2}dxdy<\mathrm{\infty },$

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

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

is called a Hilbert–Schmidt integral operator.Template:SfnTemplate:Sfn T is a Hilbert–Schmidt operator with Hilbert–Schmidt norm

$\Vert T{\Vert }_{\mathrm{H}\mathrm{S}}=\Vert k{\Vert }_{L^2}.$

Hilbert–Schmidt integral operators are both continuous and compact.Template:Sfn

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

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

is compact. If

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

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.Template:Sfn

See also

• Hilbert–Schmidt operator