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 :\int_{\Omega} \int_{\Omega} | k(x, y) |^{2} ,dx , dy is called a Hilbert–Schmidt kernel. The associated integral operator T : L2(Ω) → L2(Ω) given by :(Tf) (x) = \int_{\Omega} k(x, y) f(y) , dy is called a Hilbert–Schmidt integral operator. T is a Hilbert–Schmidt operator with Hilbert–Schmidt norm

:\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 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) = \int_{X} 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.